Trigonometric Equations

Types of trigonometric equations

1) Basic Equations 

 

A) sin x = a

If sin x = a and α is a solution of the equation then :

1. x = α + 2kπ, k E Z      or
2. x = 180° – α + 2kπ, k E Z

sin x = a –> {x = arcsin a + 2kπ, k E Z}                                                                                                                      {x = π – arcsin a + 2kπ, k E Z}

• Special Results 

1)  sin x = 0	2) sin x = 1	3) sin x = -1
x = kπ, k E Z	x = 90° + kπ, k E Z	x = 270° + 2kπ, k E Z

B) cos x = a

If con x = a and α is a solution of the equation, then :

1. x = α + 2kπ, k E Z     or
2. x = -α + 2kπ, k E Z

cos x = a –> x +- arccos a + 2kπ, k E Z

• Special Results 

1)  cos x = 0	2) cos x = 1	3) cos x = -1
x =90° + kπ, k E Z	x = 360° + kπ, k E Z	x = 180° + 2kπ, k E Z

C) tan x = a

If tan x = a and α is a solution of the equation then :

x = α + kπ, k E Z

D) cot x = a

If cot x = a and α is a solution of the equation then :

x = α + kπ, k E Z

2) Factorizing Equations 

3) Equations with a common Ratio

1. If cos x = cos a then —> x = a + 2kπ, k E Z
2. If sin x = sin a then —> x = a + 2kπ or x = (π – a) + 2kπ, k E Z
3. If tan x = tan a then —> x = a + kπ, k E Z
4. If cot x = cot a then —> x = a + kπ, k E Z

4) Homogenous Equation in sin x and cos x

If the degree of all terms in an equation are the same, the equation is called a homogenous equation.

ax + by = 0

ax 2 + bxy + cy 2 = o