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Double Angle Calculator

Select the angle type and enter its value. The calculator will provide you with the double angle identities for it.

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Double Angle Calculator

This online double angle calculator computes all basic double angle identities for a given angle. It calculates sin 2θ, cos 2θ, and tan 2θ in degrees, radians, or π radians.

What is a Double Angle?

A double angle is an angle multiplied by 2. In trigonometry, double angle formulas relate the sine, cosine, and tangent of 2θ to functions of θ.

Examples:

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

Double Angle Formulas

Sine: sin(2θ) = 2 sin(θ) cos(θ)
Cosine:
- cos(2θ) = cos²(θ) − sin²(θ)
- cos(2θ) = 2 cos²(θ) − 1
- cos(2θ) = 1 − 2 sin²(θ)
Tangent: tan(2θ) = 2 tan(θ) / (1 − tan²(θ))

How to Calculate a Double Angle

Determine the values of the basic trigonometric functions for θ.
Substitute θ into the double angle formulas above.
Compute the result.

Example: Find sin(60°) using the double angle formula.

Use the sine formula: sin(2θ) = 2 sin(θ) cos(θ)
Set θ = 30° → sin(60°) = 2 sin(30°) cos(30°)
sin(30°) = 1/2, cos(30°) = √3/2
sin(60°) = 2 × (1/2) × (√3/2) = √3/2

Deriving Double Angle Formulas

Sine Double Angle

Using the angle sum identity: sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

Set x = y = θ:

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

Cosine Double Angle

Using the angle sum identity: cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

Set x = y = θ:

cos(2θ) = cos²(θ) − sin²(θ)

Using sin²(θ) + cos²(θ) = 1, we also get:

cos(2θ) = 2 cos²(θ) − 1
cos(2θ) = 1 − 2 sin²(θ)

Tangent Double Angle

From tan(x + y) = (tan(x) + tan(y)) / (1 − tan(x) tan(y))

Set x = y = θ:

tan(2θ) = (tan(θ) + tan(θ)) / (1 − tan²(θ)) = 2 tan(θ) / (1 − tan²(θ))

How the Double Angle Calculator Works

Input:

Select the unit (degree, radian, or π radian)
Enter the angle value
Click "Calculate"

Output:

Displays sin(2θ), cos(2θ), and tan(2θ)
Step-by-step solution for verification
Option to recalculate for different angles

FAQs

Why are trigonometric identities important?

They are universally true and help solve geometry and trigonometry problems efficiently. Memorizing key identities reduces errors and speeds up calculations.

What are the six trigonometric ratios?

The six ratios are sin, cos, tan, csc, sec, and cot. They describe the relationships between the sides of a right triangle for a given angle θ.

Wrap Up

Use this double angle calculator to accurately compute sin 2θ, cos 2θ, and tan 2θ. It is fast, reliable, and works for angles in degrees, radians, or π radians.

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