Half-Angle Notes

Published on: September 29, 2026

This Half-Angle Calculator works out the sine, cosine or tangent of an angle that has been halved, and shows the working line by line. Every half-angle identity leans on the cosine of the full angle, so that value is found first and placed inside a square root. The fraction underneath the root is simplified before the root is taken. Following the same pattern every time keeps the plus and minus signs in the three identities straight.

Step-by-step method

  1. Set up the problem.
  2. Write the half-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace the cosine value with its number.
  5. Simplify the fraction inside the square root.
  6. Take the square root to get the value.

Formulas:

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

Problem

\(\cos\left(\frac{\theta}{2}\right),\; \theta = 120^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the half-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace the cosine value with its number.
  5. Simplify the fraction inside the square root.
  6. Take the square root to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\cos\left(\frac{\theta}{2}\right)\) is needed, and the full angle is \(\theta = 120^{\circ}\).

\(\theta = 120^{\circ}\)

Step 2

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

In this problem: The cosine of a halved angle is rewritten using the cosine of the full angle, giving \(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}\).

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

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(120^{\circ}\) gives \(\sqrt{\frac{1 + \cos(120^{\circ})}{2}}\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(120^{\circ})}{2}}\)

Step 4

Step 4 - Replace the cosine value with its number.

In this problem: The cosine of \(120^{\circ}\) is \(-0.5\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + -0.5}{2}}\)

Step 5

Step 5 - Simplify the fraction inside the square root.

In this problem: Adding \(-0.5\) to \(1\) and halving gives \(0.25\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{0.25}\)

Step 6

Step 6 - Take the square root to get the value.

In this problem: Taking the square root of \(0.25\) gives \(0.5\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{0.25} = 0.5\)

Final Answer

\(\cos\left(\frac{\theta}{2}\right) = 0.5\)
See Example 2 Hide Example 2

Problem

\(\sin\left(\frac{\theta}{2}\right),\; \theta = 90^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the half-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace the cosine value with its number.
  5. Simplify the fraction inside the square root.
  6. Take the square root to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\sin\left(\frac{\theta}{2}\right)\) is needed, and the full angle is \(\theta = 90^{\circ}\).

\(\theta = 90^{\circ}\)

Step 2

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

In this problem: The sine of a halved angle is rewritten using the cosine of the full angle, giving \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}\).

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

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(90^{\circ}\) gives \(\sqrt{\frac{1 - \cos(90^{\circ})}{2}}\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(90^{\circ})}{2}}\)

Step 4

Step 4 - Replace the cosine value with its number.

In this problem: The cosine of \(90^{\circ}\) is \(0\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0}{2}}\)

Step 5

Step 5 - Simplify the fraction inside the square root.

In this problem: Taking \(0\) away from \(1\) and halving gives \(0.5\).

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

Step 6

Step 6 - Take the square root to get the value.

In this problem: Taking the square root of \(0.5\) gives \(0.71\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{0.5} \approx 0.71\)

Final Answer

\(\sin\left(\frac{\theta}{2}\right) \approx 0.71\)
See Example 3 Hide Example 3

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the half-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace the cosine value with its number.
  5. Simplify the fraction inside the square root.
  6. Take the square root to get the value.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

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

In this problem: The sine of a halved angle is rewritten using the cosine of the full angle, giving \(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}\).

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

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(60^{\circ}\) gives \(\sqrt{\frac{1 - \cos(60^{\circ})}{2}}\).

\(\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(60^{\circ})}{2}}\)

Step 4

Step 4 - Replace the cosine value with its number.

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

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

Step 5

Step 5 - Simplify the fraction inside the square root.

In this problem: Taking \(0.5\) away from \(1\) and halving gives \(0.25\).

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

Step 6

Step 6 - Take the square root to get the value.

In this problem: Taking the square root of \(0.25\) gives \(0.5\).

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

Final Answer

\(\sin\left(\frac{\theta}{2}\right) = 0.5\)
See Example 4 Hide Example 4

Problem

\(\cos\left(\frac{\theta}{2}\right),\; \theta = 90^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the half-angle identity for the chosen function.
  3. Substitute the angle into the identity.
  4. Replace the cosine value with its number.
  5. Simplify the fraction inside the square root.
  6. Take the square root to get the value.

Step 1

Step 1 - Set up the problem.

In this problem: The value of \(\cos\left(\frac{\theta}{2}\right)\) is needed, and the full angle is \(\theta = 90^{\circ}\).

\(\theta = 90^{\circ}\)

Step 2

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

In this problem: The cosine of a halved angle is rewritten using the cosine of the full angle, giving \(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}\).

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

Step 3

Step 3 - Substitute the angle into the identity.

In this problem: Replacing \(\theta\) with \(90^{\circ}\) gives \(\sqrt{\frac{1 + \cos(90^{\circ})}{2}}\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(90^{\circ})}{2}}\)

Step 4

Step 4 - Replace the cosine value with its number.

In this problem: The cosine of \(90^{\circ}\) is \(0\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + 0}{2}}\)

Step 5

Step 5 - Simplify the fraction inside the square root.

In this problem: Adding \(0\) to \(1\) and halving gives \(0.5\).

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

Step 6

Step 6 - Take the square root to get the value.

In this problem: Taking the square root of \(0.5\) gives \(0.71\).

\(\cos\left(\frac{\theta}{2}\right) = \sqrt{0.5} \approx 0.71\)

Final Answer

\(\cos\left(\frac{\theta}{2}\right) \approx 0.71\)