CBSE Class 10 Maths Chapter 8 Introduction to Trigonometry Notes

Position of a point P in the Cartesian plane with respect to co-ordinate axes is represented by the ordered pair (x, y).
Trigonometry is the science of relationships between the sides and angles of a right-angled triangle.
Trigonometric Ratios: Ratios of sides of right triangle are called trigonometric ratios.
Consider triangle ABC right-angled at B. These ratios are always defined with respect to acute angle ‘A’ or angle ‘C.
If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of an angle can be easily determined.
How to identify sides: Identify the angle with respect to which the t-ratios have to be calculated. Sides are always labelled with respect to the ‘θ’ being considered.

Let us look at both cases:

In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six t-Ratios with respect to the sides.

case I	case II
(i) sine A = $\frac { perpendicular }{ hypotenuse } =\frac { BC }{ AC }$	(i) sine C = $\frac { perpendicular }{ hypotenuse } =\frac { AB }{ AC }$
(ii) cosine A = $\frac { base }{ hypotenuse } =\frac { AB }{ AC }$	(ii) cosine C = $\frac { base }{ hypotenuse } =\frac { BC }{ AC }$
(iii) tangent A = $\frac { perpendicular }{ base } =\frac { BC }{ AB }$	(iii) tangent C = $\frac { perpendicular }{ base } =\frac { AB }{ BC }$
(iv) cosecant A = $\frac { hypotenuse }{ perpendicular } =\frac { AC }{ BC }$	(iv) cosecant C = $\frac { hypotenuse }{ perpendicular } =\frac { AC }{ AB }$
(v) secant A = $\frac { hypotenuse }{ base } =\frac { AC }{ AB }$	(v) secant C = $\frac { hypotenuse }{ base } =\frac { AC }{ BC }$
(v) cotangent A = $\frac { base }{ perpendicular } =\frac { AB }{ BC }$	(v) cotangent C = $\frac { base }{ perpendicular } =\frac { BC }{ AB }$

Note from above six relationships:

cosecant A = $\frac { 1 }{ sinA }$ , secant A = $\frac { 1 }{ cosineA }$ , cotangent A = $\frac { 1 }{ tanA }$ ,

However, it is very tedious to write full forms of t-ratios, therefore the abbreviated notations are:
sine A is sin A
cosine A is cos A
tangent A is tan A
cosecant A is cosec A
secant A is sec A
cotangent A is cot A

TRIGONOMETRIC IDENTITIES

An equation involving trigonometric ratio of angle(s) is called a trigonometric identity, if it is true for all values of the angles involved. These are:
tan θ = $\frac { sin\theta }{ cos\theta }$
cot θ = $\frac { cos\theta }{ sin\theta }$

sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1
sin θ cosec θ = 1 ⇒ cos θ sec θ = 1 ⇒ tan θ cot θ = 1

ALERT:
A t-ratio only depends upon the angle ‘θ’ and stays the same for same angle of different sized right triangles.

Value of t-ratios of specified angles:

∠A	0°	30°	45°	60°	90°
sin A	0	$\frac { 1 }{ 2 }$	$\frac { 1 }{ \sqrt { 2 } }$	$\frac { \sqrt { 3 } }{ 2 }$	1
cos A	1	$\frac { \sqrt { 3 } }{ 2 }$	$\frac { 1 }{ \sqrt { 2 } }$	$\frac { 1 }{ 2 }$	0
tan A	0	$\frac { 1 }{ \sqrt { 3 } }$	1	√3	not defined
cosec A	not defined	2	√2	$\frac { 2 }{ \sqrt { 3 } }$	1
sec A	1	$\frac { 2 }{ \sqrt { 3 } }$	√2	2	not defined
cot A	not defined	√3	1	$\frac { 1 }{ \sqrt { 3 } }$	0

The value of sin θ and cos θ can never exceed 1 (one) as opposite side is 1. Adjacent side can never be greater than hypotenuse since hypotenuse is the longest side in a right-angled ∆.

‘t-RATIOS’ OF COMPLEMENTARY ANGLES

If ∆ABC is a right-angled triangle, right-angled at B, then
∠A + ∠C = 90° [∵ ∠A + ∠B + ∠C = 180° angle-sum-property]
or ∠C = (90° – ∠A)

Thus, ∠A and ∠C are known as complementary angles and are related by the following relationships:
sin (90° -A) = cos A; cosec (90° – A) = sec A
cos (90° – A) = sin A; sec (90° – A) = cosec A
tan (90° – A) = cot A; cot (90° – A) = tan A