Double Angle Formulas

Double Angle Formulas are formulas in trigonometry to solve trigonometric functions where their angle is in the multiple of 2, i.e. in the form of (2θ). Double angle formulas are special cases of trigonometric formulas and are used to solve various types of trigonometric problems.

In this article, we explore double-angle identities, double-angle identity definitions, and double-angle identity formulas by deriving all double-angle formulas, providing insight into their importance and uses in trigonometry.

What is Double Angle Formula?

Trigonometric formulae known as "double angle identities" define the trigonometric functions of double angles in terms of the trigonometric functions of the original angles. Numerous mathematical and engineering applications benefit from these identities. The identities for the sum and difference of angles lead to the identities of double angles.

Double Angle Formulas Definition

Double angle identities are trigonometric formulae that represent the angle (θ) sine, cosine, and tangent in terms of the angle (θ) sine, cosine, and tangent.

Double Angle Formulas

The table with double angle formulas is added below:

Double Angle Identities Formulas

For sine, cosine, and tangent, the primary double angle identities are as follows:

Double Angle Formulas of Sin

sin(2θ) = 2sin(θ)cos(θ)

sin(2θ) = 2tan(θ) / [1 + tan²(θ)]​

Double Angle Formulas of Cos

cos(2θ) = cos²(θ) - sin²(θ)

cos(2θ) = 2cos²(θ) - 1

cos(2θ) = 1 - 2sin²(θ)

cos(2θ) = [1 - tan²(θ)]​ / [1 + tan²(θ)]​

Double Angle Formulas of Tan

tan(2θ) = 2tan(θ) / [1 - tan²(θ)]​

These equations define the trigonometric functions of double angles (2θ) in terms of the original angles' (θ) trigonometric functions. In a variety of mathematical and engineering situations, they are helpful in decomposing trigonometric formulas and resolving issues with double angles.

Double Angle Formulas Derivation

Trigonometric formulae known as the "double angle identities" define the trigonometric functions of twice an angle in terms of the trigonometric functions of the angle itself. I'll be obtaining the sine, cosine, and tangent double angle identities here.

Derivation of Sine Double Angle Formula

Sine Double Angle Identity:

sin(2θ) = 2 sinθ cosθ

Start with the sum-to-product identity for sine:

sin (A + B) =sin A cos B + cos A sin B

Let A = θ and B = θ

sin(2θ) = sin(θ+θ)

sin(2θ) = sinθ cosθ + cosθ sinθ

sin(2θ) = 2sinθ cosθ

Derivation of Cos Double Angle Formula

Cosine Double Angle Identity:

cos(2θ) = cos²(θ) - sin²(θ)

Start with the sum-to-product identity for cosine:

cos (A + B) = cos A cos B - sin A sin B

Let A = θ and B = θ

cos(2θ) = cos(θ+θ)

cos(2θ) = cosθcosθ - sinθ sinθ

cos(2θ) = cos²θ - sin²θ

Derivation of Tan Double Angle Formula

Tangent Double Angle Identity:

tan(2θ) = 2tanθ / [1 - tan²θ​]

Use the quotient identity for tangent:

tan(A+B) = [tan A + tan B] / [1 - tan A tan B]​

Let A=θ and B=θ

tan(2θ) = tan(θ+θ)

tan(2θ) = [tan(θ) + tan(θ)] / ​[1 - tan(θ)tan(θ)]

tan(2θ) = 2tan(θ)​ / [1 - tan²(θ)]

Read More,

• Trigonometric Identities
• Trigonometry Table
• Half Angle Formulae

Examples Using Double Angle Formulas

Example 1: Solve sin(2θ) = cos(θ) for θ

Solution:

sin(2θ) = cos(θ)

Using double angle identity for sine sin(2θ)=2sin(θ)cos(θ)), substitute:

2sin(θ)cos(θ) = cos(θ)

Now, divide both sides by cos(θ) (assuming cos⁡(θ) ≠ 0

2sin(θ) = 1

Finally, solve for θ

sin(θ) = 1 /2

This implies θ = 30° or θ = 150^°.

Example 2: Express tan(2x) in terms of tan(x):

Solution:

Using double angle identity for tangent

tan(2x) = 2tan(x) ​/ {1 - tan²(x)}

This expression provides the tangent of twice the angle x in terms of the tangent of x.

Example 3: Use double angle identities to find the exact value of sin(120°)

Solution:

sin(2θ) = sin (240°)

Using, sin (180°+ θ) = - sin(θ)

We can rewrite expression,

-sin⁡ (60°) = - √3/2

Example 4: Prove the double angle identity for sine: sin(2θ) = 2sinθcosθ.

Solution:

Starting with (LHS)

sin(2θ) = sin(θ+θ)

sin(2θ) = sinθ cosθ + cosθ sinθ

Using trigonometric identity:

sin (a + b) = sin(a)cos(b) + cos(a)sin(b)

we get:

sin (θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ)

Thus, LHS is equal to the right-hand side (RHS), and double angle identity for the sine is proved.

Practice Problems on Double Angle Formulas

Q1. Solve for sin(2θ) if sinθ = 3/5​.

Q2. Express cos(2α) in terms of cos(α) if cos(α) = -4​/7.

Q3. If tan(β) = 125​, find the value of tan(2β).

Q4. Given that sin(ϕ) = 1/2​ and ϕ is acute, determine cos(2ϕ).

Q5. Evaluate cot(2θ) if cotθ = -3/4​.