Sum and Difference Identities Calculator

This Sum and Difference Identities Calculator helps you evaluate sin(α ± β), cos(α ± β), and tan(α ± β) using the correct trigonometric identities. Choose the function, choose whether you want a sum or difference, and then enter the two angles α and β. The calculator applies the matching identity and works through the result step by step. It is a simple way to check answers, understand angle-sum and angle-difference identities, and practise basic trigonometry step by step.

Step-by-step method

1. Identify what is given.
2. Write the correct identity.
3. Substitute the values and calculate.

Formulas:

Example 1:

Step 1 - Identify what is given.

In this problem: The given angles are \(\alpha = 30^{\circ}\) and \(\beta = 45^{\circ}\).

Step 2 - Write the correct identity.

In this problem: Use the identity: \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

Step 3 - Substitute the values and calculate.

In this problem: Compute and combine the terms: \(0.35 + 0.61 \approx 0.97\).

Final answer:

Example 2:

Step 1 - Identify what is given.

In this problem: The given angles are \(\alpha = 60^{\circ}\) and \(\beta = 15^{\circ}\).

Step 2 - Write the correct identity.

In this problem: Use the identity: \(\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\).

Step 3 - Substitute the values and calculate.

In this problem: Substitute and divide: \(\frac{1.46}{1.46} = 1\).

Final answer:

See Example 1 Hide Example 1

Problem

\(\sin(\alpha + \beta),\; \alpha = 30^{\circ},\; \beta = 45^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Write the correct identity.
3. Substitute the values and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given angles are \(\alpha = 30^{\circ}\) and \(\beta = 45^{\circ}\).

\(\alpha = 30^{\circ},\; \beta = 45^{\circ}\)

Step 2

Step 2 - Write the correct identity.

In this problem: Use the identity: \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

\(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: Compute and combine the terms: \(0.35 + 0.61 \approx 0.97\).

\(\sin(\alpha + \beta) = 0.35 + 0.61 \approx 0.97\)

Final Answer

\(\sin(\alpha + \beta),\; \alpha = 30^{\circ},\; \beta = 45^{\circ} = \sin(\alpha + \beta) \approx 0.97\)

See Example 2 Hide Example 2

Problem

\(\tan(\alpha - \beta),\; \alpha = 60^{\circ},\; \beta = 15^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Write the correct identity.
3. Substitute the values and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given angles are \(\alpha = 60^{\circ}\) and \(\beta = 15^{\circ}\).

\(\alpha = 60^{\circ},\; \beta = 15^{\circ}\)

Step 2

Step 2 - Write the correct identity.

In this problem: Use the identity: \(\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\).

\(\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: Substitute and divide: \(\frac{1.46}{1.46} = 1\).

\(\tan(\alpha - \beta) = \frac{1.46}{1.46} = 1\)

Final Answer

\(\tan(\alpha - \beta) = 1\)