Trigonometry questions for CAT exams are not asked very often. This is because the CAT organizers don't give a strict list of topics, so we can't say for sure if trigonometry will be there or not. However, knowing trigonometry is important for CAT preparation, especially for geometry questions. Geometry questions make up 20% of the CAT exam.

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 2023 quantitative aptitude topic Trigonometry. 

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Trigo Basic Concepts

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: 

Ration/ Angles 0° 30° 45° 60° 90°
Sin θ 0 ½ 1/√2 √3/2 1
Cos θ 1 √3/2 1/√2 ½ 0
Tan θ 0 1/√3 1 √3 ∞ or Undefined
Cosec θ ∞ or Undefined 2 √2 2/√3 1
Sec θ 1 2/√3 √2 2 ∞ or Undefined
Cot θ ∞ or Undefined √3 1 1/√3 0

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.
Sum & Diff Formulas

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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Angles Formulas

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)
Law of Sines & Cosines

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 Sample Questions

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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