Trigonometric Ratios of Allied Angles - Practice Questions & MCQ

Allied Angles (Part 1)

Two angles are called allied if their sum or difference is a multiple of π/2   

• sin (90⁰ - θ) = cos (θ)

• cos (90⁰ - θ) = sin (θ)

• tan (90⁰ - θ) = cot (θ)

• csc (90⁰ - θ) = sec (θ)          

• sec (90⁰ - θ) = csc (θ)

• cot (90⁰ - θ) = tan (θ)

• sin (90⁰ + θ) = cos (θ)

• cos (90⁰ + θ) = - sin (θ)

• tan (90⁰ + θ) = - cot (θ)

• csc (90⁰ + θ) = sec (θ)          

• sec (90⁰ + θ) = - csc (θ)

• cot (90⁰ + θ) = - tan (θ)

Allied Angles (Part 2)

• sin (180⁰ - θ) = sin (θ)

• cos (180⁰ - θ) = - cos (θ)

• tan (180⁰ - θ) = - tan (θ)

• csc (180⁰ - θ) = csc (θ)          

• sec (180⁰ - θ) = - sec (θ)

• cot (180⁰ - θ) = - cot (θ)

• sin (180⁰ + θ) = - sin (θ)

• cos (180⁰ + θ) = - cos (θ)

• tan (180⁰ + θ) = tan (θ)

• csc (180⁰ + θ) = - csc (θ)          

• sec (180⁰ + θ) = - sec (θ)

• cot (180⁰ + θ) = cot (θ)

AID TO REMEMBER

1. All the trigonometric function of a real number of the form 2n(π/2) ± x ( n ∈ I) (i.e. an even multiple of π/2 ± x) is numerically equal to the same function of x, with sign depending on the quadrant in which terminal side of the angles lies.

        For example: cos (π + x) = cos (2(π/2) + x) = - cos (x), -ve sign chosen because (π + x) lies in 3rd quadrant and ‘cos’ is -ve in third quadrant.

2. All the trigonometric function of a real number of the form (2n + 1)π/2 ± x ( n ∈ I) (i.e. an even multiple of π/2 ± x) is numerically equal to the co-function of x, with sign depending on the quadrant in which terminal side of the angles lies.

            Note that ‘sin’ and ‘cos’ are co-functions of each other, ‘tan’ and ‘cot’ are co-function of each other and ‘sec’ and ‘cosec’ are co-function of each other.

            For example: sec (π/2 +x ) = - cosec (x), as (π/2 +x) lies in the 2nd quadrant and ‘sec’ is -ve in 2nd quadrant.