Double-Angle Calculator

This Double-Angle Calculator helps you evaluate sin(2θ), cos(2θ), and tan(2θ) using the correct double-angle identities. Choose the function you want, enter the angle θ, and the calculator will apply the matching formula step by step. It is useful for simplifying trigonometric expressions, checking answers, and understanding how double-angle identities work. It is a simple way to practise basic trigonometry and build confidence with common trig formulas.

Step-by-step method

1. Identify what is given.
2. Choose the correct double-angle identity.
3. Substitute the value of θ.
4. Compute the result.

Formulas:

Example 1:

Step 1 - Identify what is given.

In this problem: The given value is \(\theta = 30^{\circ}\).

Step 2 - Choose the correct double-angle identity.

In this problem: Use the identity: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

Step 3 - Substitute the value of θ.

In this problem: Substitute \(\theta = 30^{\circ}\): \(\sin(2\theta) = 2\sin(30^{\circ})\cos(30^{\circ})\).

Step 4 - Compute the result.

In this problem: Compute: \(2 \times 0.5 \times 0.87 \approx 0.87\).

Final answer:

Example 2:

Step 1 - Identify what is given.

In this problem: The given value is \(\theta = 15^{\circ}\).

Step 2 - Choose the correct double-angle identity.

In this problem: Use the identity: \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^{2}(\theta)}\).

Step 3 - Substitute the value of θ.

In this problem: Substitute \(\theta = 15^{\circ}\): \(\tan(2\theta) = \frac{2\tan(15^{\circ})}{1 - \tan^{2}(15^{\circ})}\).

Step 4 - Compute the result.

In this problem: Compute: \(\tan(\theta) \approx 0.27\), so \(\tan(2\theta) = \frac{0.54}{0.93} \approx 0.58\).

Final answer:

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

1. Identify what is given.
2. Choose the correct double-angle identity.
3. Substitute the value of θ.
4. Compute the result.

Step 1

Step 1 - Identify what is given.

In this problem: The given value is \(\theta = 30^{\circ}\).

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

Step 2

Step 2 - Choose the correct double-angle identity.

In this problem: Use the identity: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

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

Step 3

Step 3 - Substitute the value of θ.

In this problem: Substitute \(\theta = 30^{\circ}\): \(\sin(2\theta) = 2\sin(30^{\circ})\cos(30^{\circ})\).

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

Step 4

Step 4 - Compute the result.

In this problem: Compute: \(2 \times 0.5 \times 0.87 \approx 0.87\).

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

Final Answer

\(\sin(2\theta) \text{ when } \theta = 30^{\circ} = \sin(2\theta) \approx 0.87\)

See Example 2 Hide Example 2

Problem

\(\tan(2\theta) \text{ when } \theta = 15^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Choose the correct double-angle identity.
3. Substitute the value of θ.
4. Compute the result.

Step 1

Step 1 - Identify what is given.

In this problem: The given value is \(\theta = 15^{\circ}\).

\(\theta = 15^{\circ}\)

Step 2

Step 2 - Choose the correct double-angle identity.

In this problem: Use the identity: \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^{2}(\theta)}\).

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

Step 3

Step 3 - Substitute the value of θ.

In this problem: Substitute \(\theta = 15^{\circ}\): \(\tan(2\theta) = \frac{2\tan(15^{\circ})}{1 - \tan^{2}(15^{\circ})}\).

\(\tan(2\theta) = \frac{2\tan(15^{\circ})}{1 - \tan^{2}(15^{\circ})}\)

Step 4

Step 4 - Compute the result.

In this problem: Compute: \(\tan(\theta) \approx 0.27\), so \(\tan(2\theta) = \frac{0.54}{0.93} \approx 0.58\).

\(\tan(2\theta) = \frac{0.54}{0.93} \approx 0.58\)

Final Answer

\(\tan(2\theta) \text{ when } \theta = 15^{\circ} = \tan(2\theta) \approx 0.58\)