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Double Angle Calculator

This double angle calculator finds sin 2θ, cos 2θ and tan 2θ and the half-angle values. Example: for θ = 30°, sin 2θ = 0.866025, cos 2θ = 0.5 and sin(θ/2) = sin 15° = 0.258819.

Source: Wikipedia – List of trigonometric identities. Updated: .

Any number, also negative or above 360°.
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Result

Result
Double angle 2θ = 60°: sin 0.866025 · cos 0.5 · tan 1.73205 | half angle θ/2 = 15°: sin 0.258819 · cos 0.965926 · tan 0.267949
All values
QuantityValue
Angle θ30°
sin θ0.5
cos θ0.866025
tan θ0.57735
sin 2θ0.866025
cos 2θ0.5
tan 2θ1.73205
sin(θ/2)0.258819
cos(θ/2)0.965926
tan(θ/2)0.267949
Calculation steps
  • Basic values: sin θ = 0.5, cos θ = 0.866025, tan θ = 0.57735.
  • sin 2θ = 2 · sin θ · cos θ = 2 · 0.5 · 0.866025 = 0.866025.
  • cos 2θ = cos²θ − sin²θ = 0.866025² − 0.5² = 0.5; likewise 2cos²θ − 1 = 0.5 and 1 − 2sin²θ = 0.5.
  • tan 2θ = 2 · tan θ / (1 − tan²θ) = 1.73205.
  • sin(θ/2) = √((1 − cos θ) / 2) = √((1 − 0.866025) / 2) = 0.258819.
  • cos(θ/2) = √((1 + cos θ) / 2) = √((1 + 0.866025) / 2) = 0.965926.
  • tan(θ/2) = sin θ / (1 + cos θ) = 0.5 / (1 + 0.866025) = 0.267949.
The sign of the half-angle values depends on the quadrant of θ/2, not of θ. The three forms of cos 2θ always give the same result.

How it is calculated

The double angle formulas give the sine, cosine and tangent of twice an angle from the values of the single angle; the half-angle formulas go the other way.

The formulas

Double angle: sin 2θ = 2 · sin θ · cos θ, cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ and tan 2θ = 2 · tan θ / (1 − tan²θ). Half angle: sin(θ/2) = ±√((1 − cos θ) / 2), cos(θ/2) = ±√((1 + cos θ) / 2) and tan(θ/2) = sin θ / (1 + cos θ). The quadrant of θ/2 decides the sign: sine is positive in quadrants I and II, cosine in I and IV.

Worked example

For θ = 30°, sin θ = 0.5, cos θ = 0.866025 and tan θ = 0.57735. It follows that sin 60° = 2 · 0.5 · 0.866025 = 0.866025, cos 60° = 0.866025² − 0.5² = 0.75 − 0.25 = 0.5 and tan 60° = 2 · 0.57735 / (1 − 1/3) = 1.73205. For the half angle θ/2 = 15°, sin 15° = √((1 − 0.866025) / 2) = 0.258819, cos 15° = √((1 + 0.866025) / 2) = 0.965926 and tan 15° = 0.5 / 1.866025 = 0.267949.

Frequently asked questions

What is the double angle formula for sine?

sin 2θ = 2 · sin θ · cos θ. For 30° that is 2 · 0.5 · 0.866025 = 0.866025.

Which formulas exist for cos 2θ?

Three equivalent ones: cos²θ − sin²θ, 2cos²θ − 1 and 1 − 2sin²θ. For 30° each gives 0.5.

How do you choose the sign in the half-angle formulas?

By the quadrant of θ/2. If θ/2 lies between 0° and 90°, sine and cosine are positive.

Sources and legal basis

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