Trigonometric Equations

Trigonometry is a branch of mathematics that deals with right-angled triangles. There are two major trigonometric functions: sine and cosine. Every other trigonometric function can be derived from these two.

Sine = adjacent side / hypotenuse
Cosine = base / hypotenuse
Tangent = sine / cosine
Cosecant = 1 / sine
Secant = 1 / cosine
Cotangent = 1 / tangent

There are several Pythagoras formulae and identities in trigonometry (check table of trigonometry). We will learn about trigonometric formulae, which are also known as trigonometric equations. 

Trigonometric functions according to quadrants

First Quadrant:

• sin (π/2 – θ) = cos θ
• cos (π/2 – θ) = sin θ
• sin (π/2 + θ) = cos θ
• cos (π/2 + θ) = – sin θ

Second Quadrant:

• sin (3π/2 – θ) = – cos θ
• cos (3π/2 – θ) = – sin θ
• sin (3π/2 + θ) = – cos θ
• cos (3π/2 + θ) = sin θ

Third Quadrant:

• sin (π – θ) = sin θ
• cos (π – θ) = – cos θ
• sin (π + θ) = – sin θ
• cos (π + θ) = – cos θ

Fourth Quadrant:

• sin (2π – θ) = – sin θ
• cos (2π – θ) = cos θ
• sin (2π + θ) = sin θ
• cos (2π + θ) = cos θ

Trigonometric equations involving angles

• sin(90° − x) = cos x
• cos(90° − x) = sin x
• tan(90° − x) = cot x
• cot(90° − x) = tan x
• sec(90° − x) = cosec x
• cosec(90° − x) = sec x

Trigonometric equations for sum and difference of angles

• sin(x + y) = sin(x) cos (y) + cos(x) sin(y)
• cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
• tan(x + y) = (tan x + tan y)/(1 - tan x • tan y)
• sin(x – y) = sin(x) cos(y) - cos(x) sin(y)
• cos(x – y) = cos(x) cos(y) + sin(x) sin(y)
• tan(x − y) = (tan x - tan y)/(1 + tan x • tan y)

Trigonometric equations for half angles

• sin (x/2) = ±√[(1 - cos x)/2]
• cos (x/2) = ± √[(1 + cos x)/2]
• tan (x/2) = ±√[(1 - cos x)/(1 + cos x)]
• tan (x/2) = ±√[(1 - cos x)(1 - cos x)/(1 + cos x)(1 - cos x)]
• tan (x/2) = ±√[(1 - cos x)²/(1 - cos²x)]
• tan (x/2) = (1 - cos x)/sin x

Trigonometric equations for double angles

• sin (2x) = 2 sin(x) • cos(x) = [2 tan x/(1 + tan² x)]
• cos (2x) = cos²(x) - sin²(x) = [(1 - tan² x)/(1 + tan² x)]
• cos (2x) = 2 cos²(x) - 1 = 1 - 2 sin²(x)
• tan (2x) = [2 tan(x)]/ [1 - tan²(x)]
• sec (2x) = sec² x/(2 - sec² x)
• cosec (2x) = (sec x • cosec x)/2

Trigonometric equations for triple angles

• sin 3x = 3 sin x - 4 sin³x
• cos 3x = 4 cos³x - 3 cos x
• tan 3x = [3 tanx - tan³x]/[1 - 3 tan²x]

Trigonometric equations for product of functions

• sin x ⋅ cos y = [sin(x + y) + sin(x − y)]/2
• cos x ⋅ cos y = [cos(x + y) + cos(x − y)]/2
• sin x ⋅ sin y = [cos(x − y) − cos(x + y)]/2

Trigonometric equations for sum of functions

• sin x + sin y = 2 [sin((x + y)/2) cos((x − y)/2)]
• sin x – sin y = 2 [cos((x + y)/2) sin((x − y)/2)]
• cos x + cos y = 2 [cos((x + y)/2) cos((x − y)/2)]
• cos x – cos y = −2 [sin((x + y)/2) sin((x − y)/2)]

Trigonometric equations for inverse functions

• sin⁻¹ (-x) = -sin⁻¹ x
• cos⁻¹ (-x) = π - cos⁻¹ x
• tan⁻¹ (-x) = -tan⁻¹ x
• cosec⁻¹ (-x) = -cosec⁻¹ x
• sec⁻¹ (-x) = π - sec⁻¹ x
• cot⁻¹ (-x) = π - cot⁻¹ x