Concepts with Solved Examples Trigonometry - Quantitative Aptitude (Quant)

Angles and their Relationship

• Angles are measured in several units: degrees, minutes, seconds, radians, and gradians.
• Conversions and basic relations: 1 degree = 60 minutes, 1 minute = 60 seconds.

The principal relation between degrees and radians is

π radians = 180° = 200g

Hence 1 radian = 180°/π and 1 degree = π/180 radians.

Basic Trigonometric Ratios

• In a right triangle ABC, if θ be the angle between AC & BC.
  
• If θ is one of the angle other then right angle, then the side opposite to the angle is perpendicular (P) and the sides containing the angle are taken as Base ( B) and the hypotenuse (H). In this type of triangles, we can have six types of ratios. These ratios are called trigonometric ratios.
  

Important Formulae and Identities

Pythagorean Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Quotient and Reciprocal Relations

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ
• sec θ = 1 / cos θ, cosec θ = 1 / sin θ

Range of Values of Ratios

If 0° ≤ θ ≤ 360° (one full revolution), then the ranges are:

1. -1 ≤ sin θ ≤ 1
2. -1 ≤ cos θ ≤ 1
3. -∞ < tan θ < ∞
4. -∞ < cot θ < ∞
5. sec θ ≤ -1 or sec θ ≥ 1
6. cosec θ ≤ -1 or cosec θ ≥ 1

Sign of Trigonometric Ratios (Quadrants)

The plane around the origin is divided into four quadrants, each spanning 90°. The signs of trigonometric ratios change depending on the quadrant:

• In the first quadrant (0° to 90°) all six ratios are positive.
• In the second quadrant (90° to 180°) sin and cosec are positive; cos, sec, tan, cot are negative.
• In the third quadrant (180° to 270°) tan and cot are positive; sin, cosec, cos, sec are negative.
• In the fourth quadrant (270° to 360°) cos and sec are positive; sin, cosec, tan, cot are negative.

Co-function and Even-Odd Identities

• sin(90° - θ) = cos θ and cos(90° - θ) = sin θ
• tan(90° - θ) = cot θ and cot(90° - θ) = tan θ
• sin(-θ) = -sin θ (odd), cos(-θ) = cos θ (even)
• tan(-θ) = -tan θ, cot(-θ) = -cot θ

Compound-Angle and Multiple-Angle Formulae (Key Results)

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
• sin 2A = 2 sin A cos A, cos 2A = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A
• tan 2A = 2 tan A / (1 - tan²A)

Solved Examples

Example.1 Simplify.

Sol. Write each trigonometric expression in terms of sin θ and cos θ.
Use Pythagorean and reciprocal identities to reduce to simplest form.

Example.2 If cot A  ,find the value of 3 cos A + 4 sin A, where A is in the first quadrant. 

Sol. Express cot A in terms of sin A and cos A and find sin A and cos A using the identity sin²A + cos²A = 1.
Compute 3 cos A + 4 sin A by substituting the determined values.

Example.3 Find the value of 

Sol. 

= 1 + 1 - 1 = 1

Table: Values of trigonometric ratios for some special angles.

Properties of a Triangle (Using Trigonometry)

➢ Sine Rule

• In any triangle ABC if AB, BC, AC be represented by c, a, b respectively
  
  

➢ Cosine Rule

• In a triangle ABC of having sides of any size, we have the following rule:
  
  
  
  

 ➢  Area of Triangle

• Δ = 1/2 · bc · sin A
• Δ = 1/2 · ca · sin B
• Δ = 1/2 · ab · sin C
• Δ = (abc) / (4R), where R is circumradius
• Δ = √[s(s - a)(s - b)(s - c)], where s = (a + b + c)/2 (Heron's formula)