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# 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

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

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

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

• 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

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