Multiple Angle Formulas

Trigonometry is one of the important topics in mathematics that is used in various fields. The trigonometric formulae are applied and used in various formulae, derivations, etc. This article is about the multiple angle formulae in trigonometry where we find sine, cosine, and tangent for multiple angles. This formula can easily evaluate the multiple angles for any given problem. The trigonometric functions with multiple angles are called the multiple-angle formulas. Double, half and triple angles are present under multiple angles.

Multiple Angle Formulae

When trigonometric functions with multiple angles are called multiple-angle formulas. Double, triple and half-angle formulae are under multiple-angle formulae.

• Half Angle Formulae: Consists of the half angle of trigonometric functions.

• Double Angle Formulae: Consists of the double angle of trigonometric functions.

• Triple Angle Formulae: Consists of the triple angle of trigonometric functions.

Some of the multiple-angle formulas in trigonometry are shown in the image added below:

List of Multiple Angle Formulae

Various multiple-angle formulas are,

Half Angle Formulae:

• sin x = 2 sin(x/2)cos(x/2) = (2 tan (x/2))/(1 + tan²(x/2))
• cos x = cos²(x/2) - sin²(x/2) = 2cos²(x/2) - 1 = 1 - 2sin²(x/2) = (1 - tan²(x/2))/(1 + tan²(x/2))
• tan x = (2 tan(x/2))/(1 - tan²(x/2))
• sin(x/2) = √((1 - cos x)/2)
• cos(x/2) = √((1 + cos x)/2)
• tan(x/2) = √((1 - cos x)/(1 + cos x))

Double Angle Formulae:

• sin 2x = 2sin x cos x = (2 tan x)/(1 + tan ²x) 
• cos 2x = cos ²x - sin ²x = 2cos ²x - 1 = 1 - 2sin ²x = (1 - tan ²x)/(1 + tan ²x)
• tan 2x = (2 tan x)/(1 - tan ²x) 

Triple Angle Formulae:

• sin 3x = 3sin x - 4sin³x
• cos 3x = 4cos³x - 3cos x
• tan 3x = (3tan x - tan³x)/(1 - 3tan²x)

Generalized Multiple Angle Formulae

• \sin n \theta=\sum^n_{k=0}\cos^k\theta\sin^{n-k}\theta\sin\left[\dfrac{1}{2}(n-k)\right]\pi
• \cos n \theta=\sum^n_{k=0}\cos^k\theta\sin^{n-k}\theta\sin\left[\dfrac{1}{2}(n-k)\pi\right]
• \tan n\theta=\dfrac{\sin n\theta}{\cos n \theta}

Sample Problems on Multiple Angle Formulas

Problem 1: Evaluate the following in the terms of tan: (i) sin 8x and (ii) cos 6x.

Solution:

(i) sin 8x

= sin (2(4x)) [Since, sin 2x = 2tan x /(1 + tan²x)]

= (2tan 4x)/(1+tan²(4x))

(ii) cos 6x

= cos(2(3x)) [Since, cos 2x = (1 - tan²x)/(1 + tan²x)]

= (1 - tan²3x)/(1 + tan²3x)

Problem 2: Evaluate the following in terms of its half-angle: (i) sin 6x, (ii) cos 8x, and (iii) tan 12x.

Solution:

(i) sin 6x

• sin 6x = 2 sin 3x cos 3x [Using half angle formulae]

• sin 6x = 2tan 3x /(1 + tan²3x)

(ii) cos 8x

• cos 8x = cos²4x - sin²4x [Using half angle formulae]

• cos 8x = 1 - 2sin²4x

• cos 8x = 2cos²4x -1

• cos 8x = (1 - tan²4x)/(1 + tan²4x)

(iii) tan 12x

• tan 12x = (2tan 6x) / (1 + tan²6x)

Problem 3: Find cos 4x in terms of cos x.

Solution: 

cos 4x = cos (2(2x))

= 2 cos²2x - 1

= 2×(2cos²x - 1)² - 1

= 2×(4cos⁴x + 1 - 4cos²x) - 1

= 8cos⁴x + 2 - 8cos²x - 1

= 8cos⁴x - 8cos²x + 1

Problem 4: If sin 2x = 4/5, find the value of tan x.

Solution: 

sin 2x = 4/5

cos 2x = √1 - sin² 2x = √1 - (4/5)² = √1 - 16/25 = √9/25 = 3/5

As, tan x = √(1 - cos 2x)/(1 + cos 2x)

tan x = √(1 - (3/5))/(1 + (3/5))

tan x = √(2/5)/(8/5)

tan x = √1/4 = 1/2

Problem 5: If cot x = p, then find the value of sec 2x - tan 2x in terms of p.

Solution: 

sec 2x - tan 2x = (1/cos 2x) - (sin 2x/cos 2x)

= (1 - sin 2x)/cos 2x [Using double angle formulae and sin²x + cos²x = 1]

= (sin²x + cos²x - 2sin x cos x)/ (cos²x - sin²x)

= (cos x - sin x)² /(cos x - sin x)(cos x + sin x)

= (cos x - sin x) /(cos x + sin x)     

Dividing both numerator and denominator by sin x

sec 2x - tan 2x = (cot x - 1)/(cot x + 1)

sec 2x - tan 2x = (p - 1)/(p + 1)

Problem 6: Find the value of 12 sin 10° - 16 sin³10°.

Solution:

12 sin 10° -16 sin³10° = 4 [3sin10° - 4sin³10°]

= 4[sin 3× 10°] [Since, sin 3A = 3sin A - 4sin³A]

= 4 sin 30°

= 4 × (1/2)

= 2

Problem 7: Prove that - (cot A + cosec A)² + 1 = cosec²(A/2).

Solution: 

LHS = (cot A + cosec A)² + 1 

= {(cos A /sin A) + (1/sin A)}² + 1

= [(1 + cos A)/ sin A]² + 1

= [2 cos²(A/2)/ 2 sin(A/2) cos(A/2)]² + 1

= [cos (A/2) / sin(A/2)]² +1

= cot²(A/2) + 1

= cosec²(A/2)