Trigonometry - Examples (with Solutions), Geometry, Quantitative Aptitude Notes | Study Quantitative Aptitude (Quant) - CAT

Angles and their Relationship

• Angles are measured in many units’ viz. degrees, minutes, seconds, radians, gradients.
• Where 1 degree = 60 minutes, 1 minute = 60 seconds,
  π radians = 180° = 200g
  ⇒ 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 

Range of Values of Ratios

If 0 ≤ θ ≤ 360^o, then the values for different trigonometric ratios will be as follows: 

1. - 1 ≤ Sin θ ≤ 1
2. - 1 ≤ cos θ ≤ 1
3. - ∞ ≤ tan θ ≤ ∞
4. - ∞ ≤ cot θ ≤ ∞
5. - ∞ ≤ Sec θ ≤ -1    &       1 ≤ Sec θ ≤ ∞
6. - ∞ ≤ Cosec θ ≤ - 1     &      1 ≤ Cosec θ ≤ ∞

Sign of Trigonometric Ratios

We divide the angle at a point (i.e. 360°) into 4 parts called quadrants. In the first quadrant all the trigonometric ratios are positive.

Important Results

Example.1 Simplify.

Sol.

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

Sol.

Example.3 Find the value of 

Sol. 

= 1 + 1 - 1 = 1

Table: Values of trigonometric Ratio for some special angles.

Properties of Triangle

➢ 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