Half Angle Identities

Learn how to find exact values of sine, cosine, and tangent for half angles using the half-angle formulas.

Advanced30 minLesson
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Definition

The half-angle identities allow us to find the exact values of trigonometric functions for half of a given angle. If we know cos⁡θ\cos\theta, we can find sin⁡θ2\sin\frac{\theta}{2}, cos⁡θ2\cos\frac{\theta}{2}, and tan⁡θ2\tan\frac{\theta}{2}.

Half-Angle Formulas

sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}
cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}
tan⁡θ2=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}
The ±\pm sign depends on the quadrant where θ2\frac{\theta}{2} lies:
  • Quadrant I: sine and cosine are both positive
  • Quadrant II: sine is positive, cosine is negative
  • Quadrant III: sine and cosine are both negative
  • Quadrant IV: sine is negative, cosine is positive

Try it now

Which formula correctly represents sin⁡θ2\sin\frac{\theta}{2}?

Worked Examples

Find the exact value of sin⁡15°\sin 15°.

1

Express as half angle

15°=30°215° = \frac{30°}{2}, so θ=30°\theta = 30° → sin⁡15°=sin⁡30°2\sin 15° = \sin\frac{30°}{2}

2

Find cos(30°)

From the unit circle: cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2} → cos⁡θ=32\cos\theta = \frac{\sqrt{3}}{2}

3

Apply half-angle formula

sin⁡θ2=±1−cos⁡θ2=±1−322\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}} = \pm\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} → =±2−322= \pm\sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}

4

Simplify

=±2−34=±2−32= \pm\sqrt{\frac{2-\sqrt{3}}{4}} = \pm\frac{\sqrt{2-\sqrt{3}}}{2} → =2−32= \frac{\sqrt{2-\sqrt{3}}}{2}

5

Determine the sign

15°15° is in Quadrant I, where sine is positive → Use ++

Common Mistakes

Forgetting to determine the correct sign

Why it's wrong: The half-angle formulas give ±\pm, and students often just use positive. The sign depends on which quadrant θ2\frac{\theta}{2} is in, not θ\theta.

Correct: Always identify the quadrant of the half angle first, then apply the appropriate sign for that quadrant.

Using the wrong formula for tangent

Why it's wrong: There are two forms of the tangent half-angle formula. Some students try to derive it from sin/cos half-angles, which is more complex.

Correct: Use tan⁡θ2=1−cos⁡θsin⁡θ\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} or sin⁡θ1+cos⁡θ\frac{\sin\theta}{1 + \cos\theta} directly.

Confusing half-angle with double-angle formulas

Why it's wrong: Half-angle formulas involve square roots; double-angle formulas do not. Students sometimes mix them up.

Correct: Half-angle: has \sqrt{} and ±\pm. Double-angle: no square root, uses 2θ2\theta.

Interactive Visual

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Right Triangle Trigonometry

θ =30°
5°45°85°
sin(θ)
0.5
Opp / Hyp
cos(θ)
√3/2
Adj / Hyp
tan(θ)
√3/3
Opp / Adj

Move the slider to change the angle and see how trigonometric ratios change.

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Practice Problems

15 problems
Problem 1 of 15
Easy

Which formula correctly represents sin⁡θ2\sin\frac{\theta}{2}?

Why It Matters

Half-angle identities are essential tools in advanced mathematics:
  • Finding exact values: Calculate precise values for angles like 15°15°, 22.5°22.5°, or π8\frac{\pi}{8} that aren't on the unit circle
  • Calculus integration: Many integrals involving sin⁡2x\sin^2 x or cos⁡2x\cos^2 x use half-angle substitutions
  • Physics and engineering: Wave analysis, signal processing, and antenna design use these formulas
  • Simplifying expressions: Convert complex trigonometric expressions into simpler forms
Without half-angle formulas, we'd be limited to only the angles we memorized on the unit circle!

Real World Applications

Signal Processing

In electronics and telecommunications, half-angle identities help analyze and synthesize waveforms.

Example:

When combining two radio signals, engineers use half-angle formulas to predict interference patterns and optimize signal strength.

1Try It Yourself

A signal analyst needs to find cos⁡15°\cos 15° to calculate a phase shift.

Use the half-angle formula to find the exact value.

Step 1: Write the mathematical expression

Start with θ=30°\theta = 30°:

Architecture and Design

Architects use half-angle calculations when designing structures with specific angular measurements.

Example:

A geodesic dome requires precise angle calculations. If a structural element meets at 45°45°, the half-angle of 22.5°22.5° determines the cut angle for supporting beams.

2Try It Yourself

A beam must be cut at half of a 60°60° angle.

Find sin⁡30°\sin 30° using the half-angle formula (verify the known value).

Step 1: Write the mathematical expression

Use θ=60°\theta = 60°:

Key Takeaways

  • 1Half-angle formulas find trig values for θ2\frac{\theta}{2} when you know cos⁡θ\cos\theta (or sin⁡θ\sin\theta)
  • 2sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}} — use −- in the radicand
  • 3cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}} — use ++ in the radicand
  • 4tan⁡θ2=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta} — no ±\pm needed
  • 5The ±\pm sign is determined by the quadrant of θ2\frac{\theta}{2}, not θ\theta

Frequently Asked Questions

Half-angle formulas involve taking a square root, which always produces a positive value. The ± accounts for the fact that the actual trig value could be negative (depending on quadrant). Double-angle formulas don't have square roots, so no ambiguity arises.
Half-angle formulas involve taking a square root, which always produces a positive value. The ± accounts for the fact that the actual trig value could be negative (depending on quadrant). Double-angle formulas don't have square roots, so no ambiguity arises.
Both forms give the same answer. Use 1−cos⁡θsin⁡θ\frac{1-\cos\theta}{\sin\theta} when you want to avoid division by a small number (when cos⁡θ≈−1\cos\theta \approx -1), and use sin⁡θ1+cos⁡θ\frac{\sin\theta}{1+\cos\theta} when cos⁡θ≈1\cos\theta \approx 1.
Yes! As long as you know the trig values for the original angle θ\theta, you can find the values for θ2\frac{\theta}{2}. This is especially useful for angles not on the standard unit circle.

Glossary

Half-angle identity
A formula that expresses a trig function of θ2\frac{\theta}{2} in terms of trig functions of θ\theta
Quadrant
One of four regions of the coordinate plane, determining the signs of trig functions
Radicand
The expression under a square root symbol
Exact value
A trigonometric value expressed with radicals rather than decimal approximations

Formula Card

Sine Half-Angle

sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}

Use minus in the radicand

Cosine Half-Angle

cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}

Use plus in the radicand

Tangent Half-Angle

tan⁡θ2=1−cos⁡θsin⁡θ\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

No plus-minus needed

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