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Introduction to Trigonometry for CAT

Trigonometry is an essential topic in CAT exam. Its questions come in combination with other geometrical concepts in the Quantitative Aptitude Section. Scoring well in Trigonometric questions means understanding the basic concepts well. This article covers all the basic concepts you would need.

• December 16, 2021

Trigonometry in the CAT exam is a broad topic. However, that doesn’t imply direct questions will appear in the exam from this topic. Mostly, trigonometric concepts appear in relation to geometric problems in the Quantitative Aptitude Section . The questions in relation to Trigonometric concepts appear in relation to other related concepts in the Quant section.

Most students often overlook these basic concepts. This causes them to forget the basics and results in losing points. Consequently, this results in an unsatisfactory overall CAT percentile. That’s why our CAT coaching has compiled this article to help you study important trigonometric basics in less than 7 minutes. In this article, we will discuss concepts, basic formulas and some examples in brief.

Basic Trigonometric Concepts

The fundamental concept in trigonometry is based on ratios of the angles in a triangle. These ratios are Trigonometric ratios.

There are six Trigonometric ratios. Each fundamental trigonometric ratio is unique.

Let’s consider the above triangle as ABC.

So, AC = Hypotenuse of the triangle

AB = Adjacent side to the Angle θ

BC = Opposite side to Angle θ

Then, the trigonometric ratios are

• Sin θ = Sine of angle θ = (Opposite side to Angle θ)/(Hypotenuse of the Triangle) = (BC/AC)
• Cos θ = Cosine of angle θ = (Adjacent side to Angle θ)/(Hypotenuse of the Triangle) = (AB/AC)
• Tan θ = Tangent of angle θ = (Opposite side to Angle θ)/(Adjacent side to Angle θ) = (BC/AB)
• Cot θ = Cotangent of angle θ = (Adjacent side to Angle θ)/(Opposite side to Angle θ) = (AB/BC)
• Sec θ = Secant of angle θ = (Hypotenuse of the Triangle)/(Adjacent side to Angle θ) = (AC/AB)
• Cosec θ = Cosecant of angle θ = (Hypotenuse of the Triangle)/(Opposite side to Angle θ) = (AC/BC)

Trigonometric ratios with respect to Angles:

These trigonometric ratios of some specific angles are widely used in numerous mathematical problems. Simultaneously, these ratios are easy to remember as well as they follow a certain discernible pattern.

Basic Identities and Formulae:

Trigonometric ratios of complementary angles:.

• Sin (90° − θ) = Cos θ
• Cos (90° − θ) = Sin θ
• Tan  (90° − θ) = Cot θ
• Cot (90° − θ) = Tan θ
• Sec (90° − θ) = Cosec θ
• Cosec (90° − θ) = Sec θ

Trigonometric Identities:

• Sin² θ + Cos² θ = 1
• Tan² θ + 1 = Sec² θ
• Cot² θ + 1 = Cosec² θ

Special Triangles’ Ratio:

There are two triangles that have the same trigonometric ratio value regardless of the length of their sides. These two triangles have sides with angles 45°-45°-90° and 30°-60°-90°.

In a 45°-45°-90° triangle, the ratio of sides, are 1:1:2 ½  respectively.

In a 60°-30°-90° triangle, the ratio of sides are 1:3 ½ :2 respectively.

Example Questions:

1.   In a triangle ABC, if Sin A, Sin B, Sin C are sines of angles A, B, C respectively, then a Sin (B − C) + b Sin ( C − A) + c Sin (A − B) =

in triangle ABC,

let’s assume that

a/Sin A = b/Sin B = c/Sin C = K

a = k Sin A,

b = k Sin B

c = k Sin C

Now consider the equation given which is

⇒ a Sin (B − C) + b Sin (C − A) + c Sin (A − B)

Substitute the values of a, b, c in the above equation

= k Sin A (Sin B Cos C −Cos B Sin C) + k Sin B (Sin C Cos A − Cos C Sin A) + k Sin C (Sin A Cos B − Cos A Sin B)

=k Sin A (Sin B Cos C − Cos B Sin C) + k Sin B (Sin C Cos A − Cos C Sin A) + k Sin C (Sin A Cos B − Cos A Sin B)

= k Sin A Sin B Cos C − k Sin A Cos B Sin C + k Sin B Sin C Cos A −k Sin B Cos C Sin A + k Sin C Sin A Cos B − k Sin C Cos A Sin B

Hence, the correct option is (a)

2.  If Cos A + Cos² A = 1 and a Sin 12 A + b Sin 10 A + c Sin 8  A + d Sin 6   A − 1 = 0. Find the value of a+(b/c)+d

Cos A = 1 − Cos 2 A

Cos A = Sin 2  A

Square on both sides

Then, Cos 2 A = Sin 4 A

1 − Sin 2 A = Sin 4 A

1 = Sin 4 A + Sin 2 A

Cube on both sides

1 3  = (Sin 4 A + Sin 2 A) 3

1 = Sin 12 A + Sin 6 A + 3Sin 8 A + 3Sin 10 A

Sin 12 A + Sin 6 A + 3Sin 8 A +3Sin 10 A − 1 = 0

on comparing with the given equation in the question,

a = 1, b = 3, c = 3, d = 1

a+(b/c)+d = 3

So, the correct option is (b)

Hope this article was helpful.

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Geometry - Trigonometry - Previous Year CAT/MBA Questions

The best way to prepare for Geometry - Trigonometry is by going through the previous year Geometry - Trigonometry XAT questions . Here we bring you all previous year Geometry - Trigonometry XAT questions along with detailed solutions.

Click here for previous year questions of other topics.

It would be best if you clear your concepts before you practice previous year Geometry - Trigonometry XAT questions.

Find the value of:

sin 6 15 ° + sin 6 75 ° + 6 sin 2 15 ° sin 2 75 ° sin 4 15 ° + sin 4 75 ° + 5 sin 2 15 ° sin 2 75 °

sin 15° + sin 75°

sin 15° cos 15°

None of the above

Answer: Option C

Let sin 2 15° = a and sin 2 75° = b

∴  sin 6 15 ° + sin 6 75 ° + 6 sin 2 15 ° sin 2 75 ° sin 4 15 ° + sin 4 75 ° + 5 sin 2 15 ° sin 2 75 °  =  a 3 + b 3 + 6 ab a 2 + b 2 + 5 ab

Now, a 3 + b 3 = (a + b) 3 - 3ab(a + b), and a 2 + b 2 = (a + b) 2 - 2ab

∴  sin 6 15 ° + sin 6 75 ° + 6 sin 2 15 ° sin 2 75 ° sin 4 15 ° + sin 4 75 ° + 5 sin 2 15 ° sin 2 75 °  =  ( a + b ) 3 - 3 ab ( a + b ) + 6 ab ( a + b ) 2 - 2 ab + 5 ab

Now, a + b = sin 2 15° + sin 2 75° = sin 2 15° + cos 2 15° = 1 [∵ Sin𝜃 = Cos(90 - 𝜃)]

∴  sin 6 15 ° + sin 6 75 ° + 6 sin 2 15 ° sin 2 75 ° sin 4 15 ° + sin 4 75 ° + 5 sin 2 15 ° sin 2 75 °  =  1 - 3 ab + 6 ab 1 - 2 ab + 5 ab  =  1 + 3 ab 1 + 3 ab  = 1

Hence, option (c).

A tall tower has its base at point K. Three points A, B and C are located at distances of 4 metres, 8 metres and 16 metres respectively from K. The angles of elevation of the top of the tower from A and C are complementary. What is the angle of elevation (in degrees) of the tower’s top from B?

We need more information to solve this.

Given the distances are : AE = 4 meters , EB = 8 meters and EC = 16 meters. Considering the length of ED = K. Given the angles DAE and angle DCE are complementary. Hence the angles are A and 90 - A. Tan(90 - A) = Cot A

tan DAE =  k 4  and tan DCE = tan  1 D A E  =  k 16

Hence  k 16  =  4 k

k = 8 meters. The angle DBE is given by

tan DBE =  k 8  = 1

Hence the angle is equal to 45 degrees.

A boat, stationed at the North of a lighthouse, is making an angle of 30° with the top of the lighthouse. Simultaneously, another boat, stationed at the East of the same lighthouse, is making an angle of 45° with the top of the lighthouse. What will be the shortest distance between these two boats? The height of the lighthouse is 300 feet. Assume both the boats are of negligible dimensions.

600√3 feet

300√3 feet

Answer: Option D

Let LM be the lighthouse and B 1 and B 2 be the positions of the two boats.

In ∆LMB 1 , Tan 30° = LM/MB 1 ⇒ MB 1  = LM√3 = 300√3

Also, in ∆LMB 2 , Tan 45° = LM/MB 2 ⇒ MB 2  = LM = 300

In ∆MB 1 B 2 ,  (B 1 B 2 ) 2 = (MB 1 ) 2  + (MB 2 ) 2 ⇒ (B 1 B 2 ) 2 = (300√3) 2  + (300) 2 ⇒ (B 1 B 2 ) 2  = 300 2  × [(√3) 2  + (1) 2 ] ⇒ (B 1 B 2 ) 2  = 300 2  × 4 ⇒ B 1 B 2 = 300 × 2 = 600

Hence, option (d).

If 5° ≤ x° ≤ 15°, then the value of sin 30° + cos x° - sin x° will be:

Between -1 and -0.5 inclusive

Between -0.5 and 0 inclusive

Between 0 and 0.5 inclusive

Between 0.5 and 1 inclusive

Answer: Option E

sin 30° + cos x° – sin x° where 5° ≤ x° ≤ 15°.

For the given range, cos x > sin x

So, cos x° – sin x° > 0

Also, sin 30° = 0.5

sin 30° + cos x° – sin x° > 0.5

So, first four options are eliminated.

Hence, option (e).

A person standing on the ground at point A saw an object at point B on the ground at a distance of 600 meters. The object started flying towards him at an angle of 30° with the ground. The person saw the object for the second time at point C flying at 30° angle with him. At point C, the object changed direction and continued flying upwards. The person saw the object for the third time when the object was directly above him. The object was flying at a constant speed of 10 kmph.

Find the angle at which the object was flying after the person saw it for the second time. You may use additional statement(s) if required.

Statement I: After changing direction the object took 3 more minutes than it had taken before. Statement II: After changing direction the object travelled an additional 200√3 meters.

Which of the following is the correct option?

Statement I alone is sufficient to find the angle but statement II is not.

Statement II alone is sufficient to find the angle but statement I is not.

Statement I and Statement II are consistent with each other.

Statement I and Statement II are inconsistent with each other.

Neither Statement I nor Statement II is sufficient to find the angle.

From the given data,

m∠CAB = 30°; m∠CBA = 30° and AB = 600 m

∴ BC = AC = 200√3 m … [By sine rule]

Statement I: After changing the direction the object took 3 more minutes than it had taken before.

The object travels 200√3 m from B to C at 10 km/hr

Thus, in 3 minutes it can travel 500 m. Hence, the object travels a total of 500 + 200√3m from C.

Thus, we know the hypotenuse CD by which we can find out the angle.

Statement II: After changing directions, the object travels 200√3 m.

Since, the object travels the same distance as before, this can only happen if the object stays on the course as before without changing any direction.

Thus, we can clearly see that the two angles from the statements are inconsistent with each other.

A person is standing at a distance of 1800 meters facing a giant clock at the top of a tower. At 5.00 p.m., he can see the tip of the minute hand of the clock at 30 degree elevation from his eye-level. Immediately, the person starts walking towards the tower. At 5.10 pm., the person noticed that the tip of the minute hand made an angle of 60 degrees with respect to his eye-level. Using three-dimensional vision, find the speed at which the person is walking. The length of the minutes hand is 200√3 meters (√3 = 1.732).

7.2 km/hour

7.5 km/hour

7.8 km/hour

8.4 km/hour

At 5.00 p.m., position of the person be P

PB = 1800 (Given)

m∠DPB = 30°

⇒ DB = 1800 tan 30° = 600√3

At 5.10 p.m. the minute hand of the clock moves by 60°

DC = CF = 200√3 m (Given)

∆FEC is 30° - 60° - 90° triangle.

So, EC = 100√3 m and EF = 300 m

DE = DC – EC = 200√3 – 100√3 = 100√3 m

FG = DB – DE = 600√3 – 100√3 = 500√3 m

In ∆AFG, m∠FAG = 60° and FG = 500√3 m

By theorem of 30°-60°-90° triangle,

BG = EF = 300 m

In ∆ABG, AG = 500 m and BG = 300 m ⇒ AB = 400 m

PA = PB – AB = 1800 – 400 = 1400 m = 1.4 km

Time taken = 10 minutes = (1/6) hours

Speed = 1.4 × 6 = 8.4 km/hr

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Cat quantitative aptitude questions | geometry questions for cat - trigonometry, cat questions | cat trigonometry questions | heights and distances.

CAT Questions / Geometry - Trigonometry / Question 3

T he question is from CAT Geometry - Trigonometry. This is about Heights and distances. We need to find out the ratio of the heights of the tower which are placed in a hexagon. Trigonometry is an important topic for CAT Preparation for the CAT Exam. Trigonometric ideas can be completely opposite, as in, some questions test lot of common sense with direction, heights and distances, while others could test you on identities. CAT exam could test one on both these fronts. Make sure you master both in CAT - Trigonometry.

Question 3 : Consider a regular hexagon ABCDEF. There are towers placed at B and D. The angle of elevation from A to the tower at B is 30 degrees, and to the top of the tower at D is 45 degrees. What is the ratio of the heights of towers at B and D?

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The question is "What is the ratio of heights?"

Hence, the answer is 1:2√3

Choice B is the correct answer.

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CAT 2019 Question Paper | Quants Slot 1

Cat previous year paper | cat quants questions | question 17.

CAT Questions / CAT 2019 Question Paper Quants-slot-1 / Question 17

H ere is a slightly tough question in CAT 2019. It combines the topics of trigonometry and algebra. It is very important to have done a solid CAT Preparation to be sure about the concepts and trigonometry formula to even attempt this question. Once you are clear with the concepts & formulas, it would be prudent to practice them from CAT previous year paper .

Question 17 : The number of the real roots of the equation 2cos(x(x + 1)) = 2 x + 2 -x is

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The question is "The number of the real roots of the equation 2cos(x(x + 1)) = 2 x + 2 -x is "

Hence, the answer is 1

Choice C is the correct answer.

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Trigonometry for Felines. Do Cats Really Understand Angles and Triangles?

Table of Contents

1. Introduction

The title question, “Does trigonometry come in cat?”, is asking if concepts of trigonometry can be applied to understand cat behavior and abilities. Trigonometry is a branch of mathematics that studies relationships between angles and lengths of sides in triangles. Some key trigonometric concepts are sine, cosine, tangent, cotangent, secant, and cosecant. These concepts describe ratios between angles and side lengths in right triangles. While trigonometry originated from the study of triangles, it can be applied more broadly to any cyclical phenomena or wave patterns. Cats exhibit many cyclical behaviors like sleeping, eating, grooming, and playing. Their agility and acrobatic abilities also rely on precise perception of angles and distances. This suggests trigonometry may provide insight into understanding cat capabilities and instincts.

This content will provide an overview of key trigonometry concepts and how they could potentially relate to cat behavior and abilities. We will explore if and how abstract mathematical principles like sine, cosine, tangent, and triangles apply to our feline friends. The goal is to satisfy the curiosity prompted by the title question and determine if trigonometry provides a meaningful framework for understanding the essence of cats.

What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. The word trigonometry comes from the Greek words trigonon (triangle) and metron (measure).

Some key definitions of trigonometry include:

Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations.
Of, relating to, or being in accordance with trigonometry.

Trigonometry allows us to calculate unknown sides or angles in a triangle if we know some of the other sides or angles. The field uses trigonometric functions like sine, cosine, and tangent to relate the angles and sides.

Trigonometry is important in many scientific and mathematical fields. It is used in:

• Engineering
• Architecture

Overall, trigonometry allows us to model and understand spatial relationships, which is critical for mapping, measurement, and more in math and science.

Cat Behavior

Cats have excellent spatial reasoning abilities that they rely on for hunting, navigation, and play. Spatial reasoning involves being able to visualize and mentally manipulate objects and their positions in space (Source 1) . For cats, this ability allows them to calculate distances when pouncing on prey, squeeze into tight spaces, climb to high vantage points, and map out environments in their mind (Source 2) .

A key component of feline spatial reasoning is their vestibular sense, which gives cats an innate sense of balance and spatial orientation. This allows them to gracefully navigate uneven terrain, make daring leaps and jumps, and always land on their feet. Cats use their highly flexible spines and spatial awareness to twist themselves mid-air and reorient during a fall so they land upright. Their spatial reasoning helps cats determine how to contort their body to fit into a box, bag, or other tight space. It also enables behaviors like squeezing through small openings and climbing narrow ledges or trees with confidence.

Spatial reasoning comes into play when cats hunt. They are able to pinpoint the location of prey using their excellent hearing and assess the distance and trajectory needed to pounce with accuracy. Cats rely on spatial memory to remember where resources like food, water, and litter boxes are located within their territory. Spatial mapping abilities allow cats to navigate back to places they have already explored and to find new paths and shortcuts. Overall, spatial reasoning is an integral part of the feline brain that enables their agility, predatory skills, navigation, and other complex behaviors.

Cats and Geometry

Cats regularly interact with shapes and objects in their environment. Their ability to recognize shapes allows them to navigate spaces and perform tasks. For example, cats can match the shape of their food bowl to find where their food is located. They are also adept at fitting into boxes, bags, and other containers of different shapes and sizes as part of their playful nature. According to research, cats can understand basic geometric shapes such as circles, squares, rectangles, and triangles in order to complete tasks requiring shape recognition.

One study at the University of Quebec tested cats’ ability to recognize shapes by training them to choose between two objects of different shapes to get a food reward. The cats were consistently able to correctly identify the shape they had been trained to select, demonstrating an understanding of basic geometric shapes (1). In another example, many cat owners have observed their cats enjoying sitting in square or rectangular boxes. The box’s angular shape seems pleasing to a cat’s geometry-attuned senses.

(1) https://www.quora.com/Can-animals-understand-true-geometric-shapes-in-their-environments

Cats and Angles

Cats have excellent vision and spatial awareness that allows them to perceive depth and distance. Their eyes are positioned on the front of their heads, giving them binocular vision that overlaps and allows for judging distances between objects (1). This ability to perceive depth helps cats navigate environments and accurately leap and land.

In addition, cats have a wide field of view of about 200 degrees, enhanced by the ability to move their heads up to 270 degrees. Their peripheral vision is excellent. This grants them a panoramic view of their surroundings and the ability to detect movement from the sides without having to fully turn their heads (2).

With their advanced vision, cats are able to understand basic geometric concepts like angles and angular relationships. When observing surroundings or tracking prey, cats can visually determine angles formed between objects and surfaces. For example, they can calculate the best angle of approach to leap onto a chair or to pounce on a toy. Cats likely use angular calculations instinctively to aid their agility and navigation.

Researchers have studied cats’ spatial cognition abilities using tests that require understanding angles and distances. Studies have shown cats can learn to visually compare angles and choose the larger or smaller one to earn a food reward (3). So while cats may not comprehend advanced trigonometry, their innate senses give them an implicit understanding of basic angles and spatial relationships.

(1) https://www.pawschicago.org/news-resources/all-about-cats/kitty-basics/cat-senses

(2) https://en.wikipedia.org/wiki/Cat_senses

(3) https://www.sciencedaily.com/releases/2020/04/200401100938.htm

Cats and Triangles

Cats can actually perceive and interact with triangles in their environment in a few key ways. According to a blog post on Cat Bandit, cats may recognize and respond to the triangular shapes in a visual scene when looking at objects or surfaces around them ( Source ). The blog shows a picture of a cat with multiple triangles overlaid on it and discusses how identifying those triangles can help improve pattern recognition abilities.

In particular, cats seem especially attuned to triangles when it comes to assessing whether they can fit into a tight, triangular space. Cats have great spatial awareness and appear to analyze shapes and openings geometrically. They seem to recognize that a triangular opening may allow them to sneak into an area, whereas a square opening would not. This shows an implicit understanding of angles and vertexes within triangular shapes.

Additionally, some cat toys and scratching posts feature triangular designs and surfaces. Cats may visually fixate on the triangular patterns and target scratching and playing on those specific areas. The triangular shapes likely stand out as distinct objects for the cat to interact with in their environment. So in summary, evidence indicates cats can perceive, recognize, and respond to triangular shapes and spaces around them.

Advanced Cat Math

Recent studies on cat numerical cognition show that cats have some basic mathematical abilities. Cats are able to perceive and differentiate between small quantities such as 1, 2, 3, or 4 objects. However, they struggle with larger quantities and exact numerical values above 4 (Moorthy, Content Study: PEG+CAT, 2014).

Cats can understand the concepts of “more” and “less” when comparing two sets of objects. A study found that cats spent more time examining a display with a larger quantity of food items vs a smaller quantity, indicating they understood more vs less. However, their math skills are limited to these simple comparative judgments (Research Shows Early Math Improvement with Home Use of … PBS Kids’ Series Peg + Cat, 2015).

While cats have rudimentary numerical and quantitative reasoning abilities, they do not possess advanced math skills. There is no evidence that cats can perform exact arithmetic, understand complex operations like multiplication or division, or grasp abstract mathematical concepts. Their math proficiency extends only to basic perception of small numerosities, relative quantities, and simple comparisons.

Applying Trig to Cats

While trigonometry is primarily used in fields like engineering and physics, we can conduct some thought experiments to imagine how it could be relevant to understanding cats. For example, we could look at how a cat determines the angle to pounce on prey. Cats have excellent depth perception and ability to judge distances, which likely involves rapid trigonometric calculations in their brains.

We can also look at how a cat perceives the angle of sunlight coming through a window. The intensity and angle of the light affects how warm and soothing an area is for a cat to nap. Trigonometric concepts like sines and cosines could model how the angle of sunlight changes over the course of a day.

When a cat jumps between surfaces at different heights and distances, trigonometry comes into play. The cat must calculate the optimal angle and force to exert. Physics calculations involving projectile motion and trigonometry can help explain how cats make these amazing leaps and judgements.

While cats don’t actively study trigonometry, their innate spatial reasoning abilities likely involve similar concepts. Thinking about how trigonometry manifests in cat behavior gives us some interesting insights into how their minds work.

In summary, while cats do not possess advanced mathematical knowledge or an understanding of complex concepts like trigonometry, they do have some basic geometric and spatial skills. Cats are able to judge distances and heights when jumping and climbing. They can locate objects and navigate environments using their spatial memory. Studies show cats understand object permanence and can mentally rotate objects and recognize shapes. So in their daily activities, cats rely on innate math abilities like estimation, measurement, and basic geometry. However, there is no evidence cats grasp abstract concepts like trigonometric functions. The title question “Does trigonometry come in cat?” is playful hyperbole – cats have rudimentary math skills but not higher-level math knowledge. While fascinating creatures, cats do not actually understand the principles of trigonometry.

Further Reading

There are many additional resources available for those interested in learning more about trigonometry, geometry, and cats:

The website Trigonometry CAT Questions: Concepts, Formulae to Prepare provides an overview of key trigonometry concepts and formulas that are helpful for CAT exam preparation.

The Geometry: Trigonometry Questions page on the 2IIM website has lots of example CAT quantitative questions involving trigonometry.

For an introduction to using trigonometry concepts with cats, check out the Introduction to Trigonometry for CAT guide from BellCAT.

There are many books and online learning platforms with extensive trigonometry resources as well. Mastering trigonometry takes practice, but the resources above provide a solid starting point.

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Trigonometry - Level-wise Practice Questions for CAT Preparation - CAT - Formulas, Tricks, Videos & Tests

Part of the course, trigonometry study material, notes for trigonometry - level-wise practice questions for cat preparation for cat, other chapters in level-wise practice questions for cat preparation, how to prepare for cat, frequently asked questions about cat examination.

• How many attempts does a CAT have? There is no limit for the CAT exam candidates to the number of attempts of giving the exam. A candidate can appear unlimited time throughout his / her entire lifespan. But the candidates should have some criteria for the candidate to hold a minimum of 50 % marks or equal to no CGPA or grade in Graduation.
• Is calculator allowed in CAT? Yes – a calculator is allowed in the CAT exam. But this does not mean that you can carry your own calculator in the exam. You will be allowed to use an on-screen calculator on your computer.
• Is GK asked in CAT? While GK or Business GK has not been part of the CAT testing, it is very important for an aspirant to be comfortable with these. Other examinations – like the XAT – have a section on GK. Barring that, the preparation for CAT does not end with giving the CAT examination.

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Untangling Disinformation

Ai fakes raise election risks as lawmakers and tech companies scramble to catch up.

Shannon Bond

Voters wait to cast their ballots on Jan. 23 in Loudon, N.H. Shortly before voting began, some voters in the state got calls from a faked version of President Biden's voice urging them not to vote, a sign of the potential that deepfakes could have on the electoral process. Tasos Katopodis/Getty Images hide caption

Voters wait to cast their ballots on Jan. 23 in Loudon, N.H. Shortly before voting began, some voters in the state got calls from a faked version of President Biden's voice urging them not to vote, a sign of the potential that deepfakes could have on the electoral process.

"What a bunch of malarkey." That's what thousands of New Hampshire voters heard last month when they received a robocall purporting to be from President Biden. The voice on the other end sounded like the president, and the catchphrase was his. But the message that Democrats shouldn't vote in the upcoming primary election didn't make sense.

"Your vote makes a difference in November, not this Tuesday," the voice said.

New Hampshire is investigating a robocall that was made to sound like Biden

It quickly emerged that the voice wasn't Biden at all. It was the product of artificial intelligence. Bloomberg reported that ElevenLabs, maker of the AI voice-cloning software believed to have made the digital voice, banned the account involved. On Tuesday, New Hampshire's attorney general said a Texas telemarketing company was behind the call and was being investigated for illegal voter suppression.

On Thursday, the Federal Communications Commission ruled robocalls using AI-generated voices illegal under federal telecoms law, opening the door to fines and lawsuits against violators.

AI-generated deepfakes are moving fast. Policymakers can't keep up

Faking a robocall is not new. But making a persuasive hoax has gotten easier, faster and cheaper thanks to generative AI tools that can create realistic images, video and audio depicting things that never happened.

As AI-generated deepfakes are being used to spread false information in elections around the world , policymakers, tech companies and governments are trying to catch up.

"We don't really think of [AI] as a free-standing threat but as more of a threat amplifier," said Dan Weiner, director of the elections and government program at the Brennan Center for Justice at New York University School of Law.

2024 elections are ripe targets for foes of democracy

He worries that AI will turbocharge efforts to discourage voters or spread bogus claims, especially in the immediate run-up to an election, when there's little time for journalists or campaigns to fact-check or debunk.

That's what appears to have happened last fall in Slovakia , just days before voters went to the polls. Faked audio seeming to show one candidate discussing rigging votes and raising the cost of beer started to spread online. His pro-Western party ended up losing to one led by a pro-Russian politician .

Because the stakes were high and the deepfake came at a critical moment, "there is a plausible case that that really did impact the outcome," Weiner said.

Worry and concern follow pro-Kremlin candidate's victory in Slovakia election

While high-profile fakes like the Biden robocall get a lot of attention, Josh Lawson, director of AI and democracy at the Aspen Institute, is focused on how AI could be used for personalized targeting.

"We are quickly advancing towards a point in the technology, likely before the election itself, when you can have real-time synthetic audio conversations," said Lawson, a former election lawyer who previously worked on elections at Facebook owner Meta.

He imagines a scenario where a bad actor deploys AI, sounding like a real person, to call a voter and give false information about their specific polling place. That could be repeated for other voters in multiple languages.

He's also worried about AI fakes targeting lower-profile elections, especially given the collapse of local news.

"The concern ... is not the big, bad deepfake of somebody at the top of the ticket, where all kinds of national press is going to be out there to verify it," Lawson said. "It's about your local mayor's race. It's about misinformation that's harder and harder for local journalists to tackle, when those local journalists exist at all. And so that's where we see synthetic media being something that will be particularly difficult for voters to navigate with candidates."

Deceiving voters, including spreading false information about when and where to vote, is already illegal under federal law. Many states prohibit false statements about candidates, endorsements or issues on the ballot.

But growing concerns about other ways that AI could warp elections are driving a raft of new legislation. While bills have been introduced in Congress, experts say states are moving faster.

In the first six weeks of this year, lawmakers in 27 states have introduced bills to regulate deepfakes in elections, according to the progressive advocacy group Public Citizen.

"There's huge momentum in the states to address this issue," Public Citizen President Robert Weissman said. "We're seeing bipartisan support ... to recognize there is no partisan interest in being subjected to deepfake fraud."

Many state-level bills focus on transparency, mandating that campaigns and candidates put disclaimers on AI-generated media. Other measures would ban deepfakes within a certain window — say 60 or 90 days before an election. Still others take aim specifically at AI-generated content in political ads.

These cautious approaches reflect the need to weigh potential harms against free speech rights.

"It is important to remember that under the First Amendment, even if something is not true, generally speaking you can't just prohibit lying for its own sake," Weiner said. "There is no truth-in-advertising rule in political advertising. You need to have solutions that are tailored to the problems the government has identified."

Just how prominent a role deepfakes end up playing in the 2024 election will help determine the shape of further regulation, Weiner said.

Meta will start labeling AI-generated images on Instagram and Facebook

Tech companies are weighing in too. Meta, YouTube and TikTok have begun requiring people to disclose when they post AI content. Meta said on Tuesday that it is working with OpenAI, Microsoft, Adobe and other companies to develop industrywide standards for AI-generated images that could be used to automatically trigger labels on platforms.

But Meta also came under fire this week from its own oversight board over its policy prohibiting what it calls "manipulated media." The board, which Meta funds through an independent trust, said the policy is "incoherent" and contains major loopholes, and it called on the company to overhaul it.

"As it stands, the policy makes little sense," said Michael McConnell, the board's co-chair. "It bans altered videos that show people saying things they do not say, but does not prohibit posts depicting an individual doing something they did not do. It only applies to video created through AI, but lets other fake content off the hook. Perhaps most worryingly, it does not cover audio fakes, which are one of the most potent forms of electoral disinformation we're seeing around the world."

The moves to develop laws and guardrails reining in AI in elections are a good start, said Lawson, but they won't stop determined bad actors.

He said voters, campaigns, lawmakers and tech platforms have to adapt, creating not just laws but social norms around the use of AI.

"We need to get to a place where things like deepfakes are looked at almost like spam. They're annoying, they happen, but they don't ruin our day," he said. "But the question is, this election, are we going to have gotten to that place?"

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Trigonometry Questions

Trigonometry questions given here involve finding the missing sides of a triangle with the help of trigonometric ratios and proving trigonometry identities. We know that trigonometry is one of the most important chapters of Class 10 Maths. Hence, solving these questions will help you to improve your problem-solving skills.

What is Trigonometry?

The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle.

The basic trigonometric ratios are defined as follows.

sine of ∠A = sin A = Side opposite to ∠A/ Hypotenuse

cosine of ∠A = cos A = Side adjacent to ∠A/ Hypotenuse

tangent of ∠A = tan A = (Side opposite to ∠A)/ (Side adjacent to ∠A)

cosecant of ∠A = cosec A = 1/sin A = Hypotenuse/ Side opposite to ∠A

secant of ∠A = sec A = 1/cos A = Hypotenuse/ Side adjacent to ∠A

cotangent of ∠A = cot A = 1/tan A = (Side adjacent to ∠A)/ (Side opposite to ∠A)

Also, tan A = sin A/cos A

cot A = cos A/sin A

Also, read: Trigonometry

Trigonometry Questions and Answers

1. From the given figure, find tan P – cot R.

From the given,

In the right triangle PQR, Q is right angle.

By Pythagoras theorem,

PR 2 = PQ 2 + QR 2

QR 2 = (13) 2 – (12) 2

= 169 – 144

tan P = QR/PQ = 5/12

cot R = QR/PQ = 5/12

So, tan P – cot R = (5/12) – (5/12) = 0

2. Prove that (sin 4 θ – cos 4 θ +1) cosec 2 θ = 2

L.H.S. = (sin 4 θ – cos 4 θ +1) cosec 2 θ

= [(sin 2 θ – cos 2 θ) (sin 2 θ + cos 2 θ) + 1] cosec 2 θ

Using the identity sin 2 A + cos 2 A = 1,

= (sin 2 θ – cos 2 θ + 1) cosec 2 θ

= [sin 2 θ – (1 – sin 2 θ) + 1] cosec 2 θ

= 2 sin 2 θ cosec 2 θ

= 2 sin 2 θ (1/sin 2 θ)

3. Prove that (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

LHS = (√3 + 1)(3 – cot 30°)

= (√3 + 1)(3 – √3)

= 3√3 – √3.√3 + 3 – √3

= 2√3 – 3 + 3

RHS = tan 3 60° – 2 sin 60°

= (√3) 3 – 2(√3/2)

= 3√3 – √3

Therefore, (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

Hence proved.

4. If tan(A + B) = √3 and tan(A – B) = 1/√3 ; 0° < A + B ≤ 90°; A > B, find A and B.

tan(A + B) = √3

tan(A + B) = tan 60°

A + B = 60°….(i)

tan(A – B) = 1/√3

tan(A – B) = tan 30°

A – B = 30°….(ii)

Adding (i) and (ii),

A + B + A – B = 60° + 30°

Substituting A = 45° in (i),

45° + B = 60°

B = 60° – 45° = 15°

Therefore, A = 45° and B = 15°.

5. If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

sin 3A = cos(A – 26°); 3A is an acute angle

cos(90° – 3A) = cos(A – 26°) {since cos(90° – A) = sin A}

⇒ 90° – 3A = A – 26

⇒ 3A + A = 90° + 26°

⇒ 4A = 116°

⇒ A = 116°/4

6. If A, B and C are interior angles of a triangle ABC, show that sin (B + C/2) = cos A/2.

We know that, for a given triangle, the sum of all the interior angles of a triangle is equal to 180°

A + B + C = 180° ….(1)

B + C = 180° – A

Dividing both sides of this equation by 2, we get;

⇒ (B + C)/2 = (180° – A)/2

⇒ (B + C)/2 = 90° – A/2

Take sin on both sides,

sin (B + C)/2 = sin (90° – A/2)

⇒ sin (B + C)/2 = cos A/2 {since sin(90° – x) = cos x}

7. If tan θ + sec θ = l, prove that sec θ = (l 2 + 1)/2l.

tan θ + sec θ = l….(i)

We know that,

sec 2 θ – tan 2 θ = 1

(sec θ – tan θ)(sec θ + tan θ) = 1

(sec θ – tan θ) l = 1 {from (i)}

sec θ – tan θ = 1/l….(ii)

tan θ + sec θ + sec θ – tan θ = l + (1/l)

2 sec θ = (l 2 + 1)l

sec θ = (l 2 + 1)/2l

8. Prove that (cos A – sin A + 1)/ (cos A + sin A – 1) = cosec A + cot A, using the identity cosec 2 A = 1 + cot 2 A.

LHS = (cos A – sin A + 1)/ (cos A + sin A – 1)

Dividing the numerator and denominator by sin A, we get;

= (cot A – 1 + cosec A)/(cot A + 1 – cosec A)

Using the identity cosec 2 A = 1 + cot 2 A ⇒ cosec 2 A – cot 2 A = 1,

= [cot A – (cosec 2 A – cot 2 A) + cosec A]/ (cot A + 1 – cosec A)

= [(cosec A + cot A) – (cosec A – cot A)(cosec A + cot A)] / (cot A + 1 – cosec A)

= cosec A + cot A

9. Prove that: (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

[Hint: Simplify LHS and RHS separately]

LHS = (cosec A – sin A)(sec A – cos A)

= (cos 2 A/sin A) (sin 2 A/cos A)

= cos A sin A….(i)

RHS = 1/(tan A + cot A)

= (sin A cos A)/ (sin 2 A + cos 2 A)

= (sin A cos A)/1

= sin A cos A….(ii)

From (i) and (ii),

i.e. (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

10. If a sin θ + b cos θ = c, prove that a cosθ – b sinθ = √(a 2 + b 2 – c 2 ).

a sin θ + b cos θ = c

Squaring on both sides,

(a sin θ + b cos θ) 2 = c 2

a 2 sin 2 θ + b 2 cos 2 θ + 2ab sin θ cos θ = c 2

a 2 (1 – cos 2 θ) + b 2 (1 – sin 2 θ) + 2ab sin θ cos θ = c 2

a 2 – a 2 cos 2 θ + b 2 – b 2 sin 2 θ + 2ab sin θ cos θ = c 2

a 2 + b 2 – c 2 = a 2 cos 2 θ + b 2 sin 2 θ – 2ab sin θ cos θ

a 2 + b 2 – c 2 = (a cos θ – b sin θ ) 2

⇒ a cos θ – b sin θ = √(a 2 + b 2 – c 2 )

Practice Questions on Trigonometry

Solve the following trigonometry problems.

• Prove that (sin α + cos α) (tan α + cot α) = sec α + cosec α.
• If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
• If sin θ + cos θ = √3, prove that tan θ + cot θ = 1.
• Evaluate: 2 tan 2 45° + cos 2 30° – sin 2 60°
• Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

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What was the first dinosaur and why do cats like to eat butter? Try our kids’ quiz

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• 1. Jacob, 7, asks: what was the first dinosaur that lived? Spinosaurus, who had a large fin, like a sail, along its back and could probably swim Tyrannosaurus rex, who was about 4 metres tall and had razor-sharp teeth Nyasasaurus parringtoni, who had a tall neck and long legs, and measured two to three metres long Diplodocus, who had a very long neck Reveal
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1. CAT Questions

A CAT Geometry question from the topic - CAT Trigoneometry that appears in the Quantitative Aptitude section of the CAT Exam broadly tests an aspirant on the concepts - Basic Trigonometric Functions, Heights and Distances, Sine rule, Cosine rule etc . In CAT Exam, one can generally expect to get approx. 1 question from CAT Trigonmetry.

2. Trigonometry for CAT 2023: Sample Questions, Basic Concepts and

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Question 1. The elevation angle of the top of the building at a distance of 50 m from its foot on a horizontal plane is found to be 60°.

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8. Introduction to Trigonometry for CAT

By bellcat December 16, 2021 Trigonometry in the CAT exam is a broad topic. However, that doesn't imply direct questions will appear in the exam from this topic. Mostly, trigonometric concepts appear in relation to geometric problems in the Quantitative Aptitude Section.

9. Previous Year CAT Questions for Geometry

Geometry - Trigonometry CAT Questions Quantitative Ability Arithmetic - Ratio, Proportion & Variation Learn Concepts Ratio, Proportion, Variation and Partnerships Concept Review Exercise CRE 1 - Ratio CRE 2 - Proportion CRE 3 - Variation CRE 4 - Partnership Practice Exercise PE 1 - Ratio PE 2 - Ratio PE 3 - Ratio PE 4 - Ratio PE 5 - Ratio

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This Trigonometry for CAT page is a collection of topic-wise notes, short techniques, tips and tricks, important formulas and topic-wise tests based on Previous Year papers to solve Trigonometry in CAT examination. This collection is designed in a way where you get a complete package for your preparation of the Trigonometry for CAT in one place ...

12. Geometry

Solution Discuss Report 2. XAT 2022 QADI | Geometry - Trigonometry XAT Question A tall tower has its base at point K. Three points A, B and C are located at distances of 4 metres, 8 metres and 16 metres respectively from K. The angles of elevation of the top of the tower from A and C are complementary.

13. CAT Questions

Method of solving this CAT Question from CAT Geometry - Trigonometry : When you look at your reflection through a mirror, the image is at a distance equal to the distance between mirror and you. Now, think about what this has to with trigonometry. Let the hexagon ABCDEF be of side 'a'. Line AD = 2a.

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H ere is a slightly tough question in CAT 2019. It combines the topics of trigonometry and algebra. It is very important to have done a solid CAT Preparation to be sure about the concepts and trigonometry formula to even attempt this question. Once you are clear with the concepts & formulas, it would be prudent to practice them from CAT previous year paper.

15. CAT Preparation -Trigonometry Question 01

CAT level question in Trigonometry - 01Questions: Sin²⁰¹⁴x + Cos²⁰¹⁴x = 1, x in the range of [ -5π, 5π ], how many values can x take?(a) 0 (b) 10(c)...

16. CAT Geometry Questions Tips & Concepts

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19. Trigonometry

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20. CAT Preparation -Trigonometry Question 02

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22. Trigonometry Questions

1. From the given figure, find tan P - cot R. Solution: From the given, PQ = 12 cm PR = 13 cm In the right triangle PQR, Q is right angle. By Pythagoras theorem, PR 2 = PQ 2 + QR 2 QR 2 = (13) 2 - (12) 2 = 169 - 144 = 25 QR = 5 cm tan P = QR/PQ = 5/12 cot R = QR/PQ = 5/12 So, tan P - cot R = (5/12) - (5/12) = 0 2.

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24. Trigonometry Practice Questions

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