Half-Angle Calculator

This Half-Angle Calculator helps you evaluate sin(θ/2), cos(θ/2), and tan(θ/2) using the correct half-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 half-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. Write the correct half-angle identity.
3. Substitute the values and calculate.

Formulas:

Example 1:

Step 1 - Identify what is given.

In this problem: The given angle is \(\theta = 60^{\circ}\).

Step 2 - Write the correct half-angle identity.

In this problem: Use the identity: \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}\).

Step 3 - Substitute the values and calculate.

In this problem: First find \(\cos(60^{\circ}) = 0.5\), then substitute and compute: \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.5}{2}} = \sqrt{0.25} = 0.5\).

Final answer:

Example 2:

Step 1 - Identify what is given.

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

Step 2 - Write the correct half-angle identity.

In this problem: Use the identity: \(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\).

Step 3 - Substitute the values and calculate.

In this problem: First find \(\cos(30^{\circ}) \approx 0.87\), then substitute and compute: \(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.87}{1 + 0.87}} = \sqrt{0.07} \approx 0.27\).

Final answer:

See Example 1 Hide Example 1

Problem

\(\sin\left(\frac{\theta}{2}\right) \text{ with } \theta = 60^{\circ}\)

Approach

Step-by-step method

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

Step 1

Step 1 - Identify what is given.

In this problem: The given angle is \(\theta = 60^{\circ}\).

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

Step 2

Step 2 - Write the correct half-angle identity.

In this problem: Use the identity: \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: First find \(\cos(60^{\circ}) = 0.5\), then substitute and compute: \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.5}{2}} = \sqrt{0.25} = 0.5\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.5}{2}} = \sqrt{0.25} = 0.5\)

Final Answer

\(\sin\left(\frac{\theta}{2}\right) = 0.5\)

See Example 2 Hide Example 2

Problem

\(\tan\left(\frac{\theta}{2}\right) \text{ with } \theta = 30^{\circ}\)

Approach

Step-by-step method

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

Step 1

Step 1 - Identify what is given.

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

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

Step 2

Step 2 - Write the correct half-angle identity.

In this problem: Use the identity: \(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\).

\(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: First find \(\cos(30^{\circ}) \approx 0.87\), then substitute and compute: \(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.87}{1 + 0.87}} = \sqrt{0.07} \approx 0.27\).

\(\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.87}{1 + 0.87}} = \sqrt{0.07} \approx 0.27\)

Final Answer

\(\tan\left(\frac{\theta}{2}\right) \text{ with } \theta = 30^{\circ} = \tan\left(\frac{\theta}{2}\right) \approx 0.27\)