# Introduction to Trigonometry

Trigonometry is that branch of Mathematics which deals with the measurement of the sides and the angles of a triangle and the problems related to angles.

### Trigonometric Ratios

Ratios of the sides of a triangle with respect to its acute angles are called trigonometric ratios.

In the right angled ∆AMP, For acute angle PAM = θ

Base = AM = x, Perpendicular = PM = y, Hypotenuse = AP = r

sin θ = y/r

cos θ = x/r

tan θ = y/x

cosec θ = r/y

sec θ = r/x

cot θ = x/y

If two sides of any right triangle are given, then all the six trigonometric ratios can be written. If one trigonometric ratio is given, then other trigonometric ratios can be written by using Pythagoras theorem or trigonometric identities.

### Trigonometric Identities

An equation involving trigonometric ratios of an angle θ is said to be a trigonometric identity if it is satisfied for all values of θ for which the given trigonometric ratios are defined.

$$ \sin^2 \theta + \cos^2 \theta = 1 $$

$$ \sec^2 \theta - \tan^2 \theta = 1 \text{ or } \sec^2 \theta = 1 + \tan^2 \theta $$

$$ \csc^2 \theta - \cot^2 \theta = 1 \text{ or } \csc^2 \theta = 1 + \cot^2 \theta $$

### T-Ratios of Complementary Angles

If θ is an acute angle, then (90º - θ) is a complementary angle for θ.

sin (90º - θ) = cos θ and cos(90º - θ) = sin θ

tan (90º - θ)= cot θ and cot (90º - θ) = tan θ

cosec (90º - θ) = sec θ and sec(90º - θ) = cosec θ

### Applications of Trigonometry

**Line of sight:** If an observer is at O and the point P is under consideration then the line OP is called line of sight of the point P.

**Angle of elevation:** Angle between the line of sight and the horizontal line OA is known as angle of elevation of point P as seen from O.

**Angle of depression:** If an observer is at P and the object under consideration is at O, then the ∠BPO is known as angle of depression of O as seen from P.

**Relation between angle of elevation and angle of depression:** Angle of elevation of a point P as seen form O is equal to the angle of depression of O as seen from P.