Double-Angle Notes

Published on: September 29, 2026

This Double-Angle Calculator works out the sine, cosine or tangent of an angle that has been doubled, and shows the working line by line. Each identity rewrites the doubled angle using values of the original angle only, so nothing beyond the ordinary trigonometric values is ever needed. Those values are substituted in and the expression is then simplified to a single number. Following the same pattern every time keeps the three identities from being mixed up.

Step-by-step method

  1. Set up the problem.
  2. Write the double-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace each trigonometric value with its number.
  5. Simplify to get the value.

Formulas:

\(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
\(\cos(2\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)\)
\(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^{2}(\theta)}\)
See Example 1 Hide Example 1

Problem

\(\cos(2\theta),\; \theta = 45^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the double-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace each trigonometric value with its number.
  5. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\cos(2\theta)\) is needed, and the angle is \(\theta = 45^{\circ}\).

\(\theta = 45^{\circ}\)

Step 2

Step 2 - Write the double-angle identity for the chosen function.

In this problem: The cosine of a doubled angle is rewritten in terms of the angle on its own, giving \(\cos(2\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)\).

\(\cos(2\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)\)

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(45^{\circ}\) gives \(\cos^{2}(45^{\circ}) - \sin^{2}(45^{\circ})\).

\(\cos(2\theta) = \cos^{2}(45^{\circ}) - \sin^{2}(45^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: The cosine of \(45^{\circ}\) is \(0.71\) and the sine is \(0.71\).

\(\cos(2\theta) = \left(0.71\right)^{2} - \left(0.71\right)^{2}\)

Step 5

Step 5 - Simplify to get the value.

In this problem: Squaring each value gives \(0.5\) and \(0.5\), and subtracting leaves \(0\).

\(\cos(2\theta) = 0.5 - 0.5 = 0\)

Final Answer

\(\cos(2\theta) = 0\)
See Example 2 Hide Example 2

Problem

\(\sin(2\theta),\; \theta = 60^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the double-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace each trigonometric value with its number.
  5. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\sin(2\theta)\) is needed, and the angle is \(\theta = 60^{\circ}\).

\(\theta = 60^{\circ}\)

Step 2

Step 2 - Write the double-angle identity for the chosen function.

In this problem: The sine of a doubled angle is rewritten in terms of the angle on its own, giving \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

\(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(60^{\circ}\) gives \(2\sin(60^{\circ})\cos(60^{\circ})\).

\(\sin(2\theta) = 2\sin(60^{\circ})\cos(60^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: The sine of \(60^{\circ}\) is \(0.87\) and the cosine is \(0.5\).

\(\sin(2\theta) = 2 \times 0.87 \times 0.5\)

Step 5

Step 5 - Simplify to get the value.

In this problem: Multiplying \(0.87\) by \(0.5\) and doubling gives \(0.87\).

\(\sin(2\theta) \approx 0.87\)

Final Answer

\(\sin(2\theta) \approx 0.87\)
See Example 3 Hide Example 3

Problem

\(\sin(2\theta),\; \theta = 30^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the double-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace each trigonometric value with its number.
  5. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\sin(2\theta)\) is needed, and the angle is \(\theta = 30^{\circ}\).

\(\theta = 30^{\circ}\)

Step 2

Step 2 - Write the double-angle identity for the chosen function.

In this problem: The sine of a doubled angle is rewritten in terms of the angle on its own, giving \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

\(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(30^{\circ}\) gives \(2\sin(30^{\circ})\cos(30^{\circ})\).

\(\sin(2\theta) = 2\sin(30^{\circ})\cos(30^{\circ})\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: The sine of \(30^{\circ}\) is \(0.5\) and the cosine is \(0.87\).

\(\sin(2\theta) = 2 \times 0.5 \times 0.87\)

Step 5

Step 5 - Simplify to get the value.

In this problem: Multiplying \(0.5\) by \(0.87\) and doubling gives \(0.87\).

\(\sin(2\theta) \approx 0.87\)

Final Answer

\(\sin(2\theta) \approx 0.87\)
See Example 4 Hide Example 4

Problem

\(\tan(2\theta),\; \theta = 30^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the double-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace each trigonometric value with its number.
  5. Simplify to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\tan(2\theta)\) is needed, and the angle is \(\theta = 30^{\circ}\).

\(\theta = 30^{\circ}\)

Step 2

Step 2 - Write the double-angle identity for the chosen function.

In this problem: The tangent of a doubled angle is rewritten in terms of the angle on its own, giving \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^{2}(\theta)}\).

\(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^{2}(\theta)}\)

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(30^{\circ}\) gives \(\frac{2\tan(30^{\circ})}{1 - \tan^{2}(30^{\circ})}\).

\(\tan(2\theta) = \frac{2\tan(30^{\circ})}{1 - \tan^{2}(30^{\circ})}\)

Step 4

Step 4 - Replace each trigonometric value with its number.

In this problem: The tangent of \(30^{\circ}\) is \(0.58\).

\(\tan(2\theta) = \frac{2 \times 0.58}{1 - \left(0.58\right)^{2}}\)

Step 5

Step 5 - Simplify to get the value.

In this problem: The top comes to \(1.15\) and the bottom to \(0.67\), and dividing gives \(1.73\).

\(\tan(2\theta) = \frac{1.15}{0.67} \approx 1.73\)

Final Answer

\(\tan(2\theta) \approx 1.73\)