Double & Half Angle Calculator

Understanding the double and half angle calculator — concept and principle

Double-angle and half-angle formulas express the trigonometric functions of 2θ and θ/2 in terms of those of θ. This calculator takes an angle θ in degrees and returns sin, cos and tan of both 2θ and θ/2. Students use it to check identity proofs, find exact values such as sin 15° or tan 22.5°, simplify integrals, and verify results in physics problems such as projectile range, which depends on sin 2θ.

How it is calculated

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• tan 2θ = 2 tan θ ÷ (1 − tan²θ)
• sin(θ/2) = ±√((1 − cos θ) ÷ 2), cos(θ/2) = ±√((1 + cos θ) ÷ 2)
• tan(θ/2) = sin θ ÷ (1 + cos θ) = (1 − cos θ) ÷ sin θ

The angle is converted to radians (θ × π ÷ 180) and the functions are evaluated directly, so the signs are always correct.

The principle

All of these follow from the angle-addition formulas sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b − sin a sin b. Setting a = b = θ gives the double-angle formulas. Solving cos 2x = 1 − 2sin²x and cos 2x = 2cos²x − 1 for sin x and cos x, then putting x = θ/2, gives the half-angle formulas. The ± sign depends on the quadrant of θ/2, which is why tan(θ/2) = sin θ ÷ (1 + cos θ) is handy: it carries the correct sign automatically.

Worked example

1. Default input: θ = 30°, so sin θ = 0.5 and cos θ = 0.8660254.
2. sin 60° = 2 × 0.5 × 0.8660254 = 0.8660254038.
3. cos 60° = 0.75 − 0.25 = 0.5.
4. tan 60° = 0.8660254 ÷ 0.5 = 1.732050808 (= √3).
5. sin 15° = √((1 − 0.8660254) ÷ 2) = √0.0669873 = 0.2588190451.
6. cos 15° = √((1 + 0.8660254) ÷ 2) = √0.9330127 = 0.9659258263.
7. tan 15° = 0.5 ÷ 1.8660254 = 0.2679491924 (= 2 − √3).

How to read the result

Exact forms are worth recognising: sin 15° = (√6 − √2)/4, cos 15° = (√6 + √2)/4 and tan 15° = 2 − √3. Doubling an angle doubles the frequency of a wave: sin 2θ completes two cycles while sin θ completes one. A value of tan 2θ that is extremely large means 2θ is near 90° or 270°, where tangent is undefined.

Common mistakes

• Writing sin 2θ = 2 sin θ; sin 60° is 0.866, not 2 × 0.5 = 1.
• Forgetting the ± in the half-angle formulas and choosing the wrong sign.
• Using the tan 2θ formula when tan θ = ±1 (θ = 45° + 90°k), where the denominator is zero.

Frequently asked questions

How do I find the exact value of sin 22.5°?

Use θ = 45°: sin 22.5° = √((1 − √2/2) ÷ 2) = √(2 − √2)/2 ≈ 0.3827.

Why does projectile range use sin 2θ?

Range on level ground is v² sin 2θ ÷ g (ignoring air resistance), which comes from 2 sin θ cos θ and is largest at θ = 45°.

What is the triple-angle formula?

sin 3θ = 3 sin θ − 4 sin³θ and cos 3θ = 4 cos³θ − 3 cos θ.

Sources: NIST Digital Library of Mathematical Functions, §4.21 (identities for trigonometric functions) · Abramowitz and Stegun, Handbook of Mathematical Functions, §4.3