- 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 =||(i) sine C =|
|(ii) cosine A =||(ii) cosine C =|
|(iii) tangent A =||(iii) tangent C =|
|(iv) cosecant A =||(iv) cosecant C =|
|(v) secant A =||(v) secant C =|
|(v) cotangent A =||(v) cotangent C =|
Note from above six relationships:
cosecant A = , secant A = , cotangent A = ,
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
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 =
cot θ =
- 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
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:
|tan A||0||1||√3||not defined|
|cosec A||not defined||2||√2||1|
|sec A||1||√2||2||not defined|
|cot A||not defined||√3||1||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