CAT Quantitative aptitude preparation should include trigonometry topics also. However, no direct questions will be asked in CAT Question Paper from the trigonometry topics but the basics will help you in solving Geometry questions which hold around 20% weightage in CAT. Check CAT Syllabus 2021

Trigonometry focuses on the relationship between the sides and the angles of a triangle. The trigonometry angles which are commonly used in trigonometry problems are  0°, 30°, 45°, 60°, and 90°. Also, there are 6 trigonometric functions that relate the angle measures of a right triangle to the length of its sides. Read the article to know more about the basics formulas, concepts, and solved questions on CAT 2021 quantitative aptitude topic Trigonometry. 

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CAT Trigonometry Basic Concepts

Based on the above triangle:

• Hypotenuse of the triangle can be represented as AC
• Perpendicular can be represented by line AB 
• Base can be represented by line BC
• AB = Side adjacent to angle ∠A
• BC = Side opposite to angle ∠A

Therefore based on above triangle we can conclude that:

• Sin A = BC/AC
• cos A = AB/AC
• tan A = BC/AB
• cosec A = 1/ sin A
• sec A = 1/ cos A
• cot AA = 1/ tan A

CAT Trigonometry Table of Angles

The important formulas related to the trigonometry angles are mentioned below: 

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Definitions and Fundamental identities of Trigonometric Functions

Fundamental Identities

Reciprocal Identities

• sin θ = 1/(cos θ) 
• csc θ = 1/(sin θ)
• cos θ = 1/(sec θ) 
• sec θ = 1/(cos θ)
• tan θ = 1/(cot θ) 
• cot θ = 1/(tan θ)

Quotient Identities

• tan θ = (sin θ)/(cos θ) 
• cot θ = (cos θ)/(sin θ) 

Pythagorean Identities

• sin²θ + cos²θ = 1
• tan²θ + 1 = sec²θ
• cot²θ + 1 = cosec²θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ – sin²θ
• tan 2θ = 2 tan θ / (1 – tan²θ)
• cot 2θ = (cot²θ – 1) / 2 cot θ

Negative Angle Identities

• sin(−θ) = − sin θ
• cos(−θ) = cos θ
• tan(−θ) = − tan θ
• csc(−θ) = − csc θ 
• sec(−θ) = sec θ 
• cot(−θ) = − cot θ

Complementary Angle Theorem

• If two acute angles add up to be 90°, they are considered complementary
• The following are considered cofunctions:
  • sine and cosine
  • tangent and cotangent 
  • secant and cosecant
• The complementary angle theorem says that cofunctions of complementary angles are equal.

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Sum and Difference Formulas in CAT Trigonometry 

Sum and difference formulas for sines and cosines: 

• sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
• cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
• sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
• cos(u – v) = cos(u)cos(v) + sin(u)sin(v)

Sum and difference formulas for tangent

• tan(u+v) = (tan(u) + tan(v))/ (1−tan(u) tan(v))
• tan(u-v) = (tan(u) − tan(v))/ (1+tan(u) tan(v))

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Double and Half Angle Formulas for CAT Trigonometry

Double Angle Formulas

• sin(2θ) = 2sin θ cos θ
• cos(2θ) = cos²θ−sin²θ = (1−2sin²θ) =  2cos²θ – 1 
• tan2θ = 2 tanθ/ (1 – tan²θ)

Half Angle Formulas

• sin (α/2) = ± √ 1− cosα / 2
• cos (α/2) = ± √ 1 + cosα / 2
• tan (α/2) = ± √ (1− cosα / 2) / 1 + cosα / 2 = sin α/ 1 + cosα = 1 - cosα/ sin α

Product to Sum Formulas

• cosαcosβ = ½ [cos(α−β)+cos(α+β)]
• sinαcosβ = ½ [sin(α+β)+sin(α−β)]
• sinαsinβ = ½ [cos(α−β)−cos(α+β)]
• cosαsinβ = ½ [sin(α+β)−sin(α−β)]

Sum to Product Formulas

• sinα+sinβ = 2sin(α+β/2)cos(α−β/2)
• sinα−sinβ = 2sin(α−β/2)cos(α+β/2)
• cosα−cosβ = −2sin(α+β/2)sin(α−β/2)
• cosα+cosβ = 2sin(α+β/2)sin(α−β/2)

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Law of Sines and Cosines

• Law of Sines = a/ sinα = b/ cosβ = c/ sinγ
• Law of Cosines = c² = a² + b² – 2ab cosγ

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CAT Trigonometry Sample Questions

Ques. Consider a regular hexagon TUVXYZ. There are towers placed at U and X. The angle of elevation from T to the tower at U is 30 degrees, and to the top of the tower at X is 45 degrees. What is the ratio of the heights of towers at U and X?

Options: 

1. 1:√3
2. 1:2√3
3. 1:2
4. 3:4√3

Ques. Find the maximum and minimum value of 8 cos A + 15 sin A + 15

Options: 

1. 11√2+15
2. 30; 8
3. 32; -2
4. 23; 8

Ques. If the sides 50 m and 130 m of the triangular field meet at an angle of 72°, then find the area in which wheat is cultivated. (sin 72° = 0.9510, cos 72° = 0.309)

Options: 

1. 100 p m 
2. 125 p m
3. 160 p m 
4. None of these

Ques. If cos A + cos2 A = 1 and a sin12 A + b sin10 A + c sin8 A + d sin6 A - 1 = 0. Find the value of a+b/c+d

Options: 

1. 4
2. 3
3. 6
4. 1

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