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Pre-Calculus Trigonometric identities

Double-angle and half-angle formulas

20 practice questions 0 video lessons Theory + worked examples

Double-Angle and Half-Angle Formulas

California Pre-Calculus • Standard F-TF.9 • Trigonometric Identities

Double-Angle and Half-Angle Formulas is a topic in Trigonometric Identities in the California Common Core State Standards. It is aligned to Standard F-TF.9, which requires students to prove and use the double-angle formulas.

The double-angle and half-angle formulas rewrite \(\sin 2\theta\), \(\cos 2\theta\), and \(\sin\dfrac{\theta}{2}\) — special cases of the sum formulas used for exact values and simplification.

California Pre-Calculus › Trigonometric Identities › Double-Angle and Half-Angle Formulas  —  Standard F-TF.9

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Theory

Setting \(A=B=\theta\) in the sum formulas gives the double-angle formulas; solving those for the half-angle gives the half-angle formulas.

\[\sin 2\theta=2\sin\theta\cos\theta,\qquad \cos 2\theta=\cos^2\theta-\sin^2\theta.\]

The cosine double angle has three equivalent forms — choose whichever uses the ratio you already know.

Half-angle sign: the \(\pm\) is decided by the quadrant of \(\dfrac{\theta}{2}\), not \(\theta\).
Double-angle formulas Double-angle formulas Double-angle formulas sin 2θ = 2 sinθ cosθ cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
The double-angle formulas, including the three forms of \(\cos 2\theta\).
Half-angle formulas Half-angle formulas Half-angle formulas sin(θ/2) = ±√((1−cosθ)/2) cos(θ/2) = ±√((1+cosθ)/2)
The half-angle formulas for sine and cosine.

Double- and half-angle formulas:

\[\sin 2\theta=2\sin\theta\cos\theta,\quad \cos 2\theta=1-2\sin^2\theta=2\cos^2\theta-1\]
\[\sin\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-\cos\theta}{2}},\quad \cos\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1+\cos\theta}{2}}\]
sine of 2 theta is 2 sine theta cosine theta; cosine of 2 theta has three forms; half-angle formulas use one plus or minus cosine over two
Pick the cosine form that matches your known value to avoid extra work.

How to use these formulas

  1. Identify whether the angle is doubled or halved.
  2. Choose the form that uses the ratio you have.
  3. Substitute and simplify.
  4. For a half-angle, set the \(\pm\) sign from the quadrant of \(\dfrac{\theta}{2}\).
Example 1 — Double angle for sine
If \(\sin\theta=\dfrac{3}{5}\) and \(\cos\theta=\dfrac{4}{5}\), find \(\sin 2\theta\).
Solution

Use \(\sin 2\theta=2\sin\theta\cos\theta\).

\(\sin 2\theta\)\(=\)\(2\cdot\dfrac{3}{5}\cdot\dfrac{4}{5}\)
\(=\)\(\dfrac{24}{25}\)
sine of 2 theta is twenty four twenty fifths
Example 2 — Double angle for cosine
With \(\sin\theta=\dfrac{3}{5}\), find \(\cos 2\theta\).
Solution

Use the version \(\cos 2\theta=1-2\sin^2\theta\) so only sine is needed.

\(\cos 2\theta\)\(=\)\(1-2\left(\dfrac{3}{5}\right)^2\)
\(=\)\(1-\dfrac{18}{25}=\dfrac{7}{25}\)
cosine of 2 theta is seven twenty fifths
Example 3 — Half angle for an exact value
Find \(\cos 15^\circ\) using the half-angle formula on \(30^\circ\).
Solution

\(15^\circ=\dfrac{30^\circ}{2}\); cosine is positive in QI, so take the \(+\) root.

\(\cos 15^\circ\)\(=\)\(\sqrt{\dfrac{1+\cos 30^\circ}{2}}\)
\(=\)\(\sqrt{\dfrac{1+\dfrac{\sqrt3}{2}}{2}}=\dfrac{\sqrt{2+\sqrt3}}{2}\)
cosine of 15 degrees is the square root of 2 plus root 3, over 2
Example 4 — Use a double-angle identity to simplify
Simplify \(2\sin 3x\cos 3x\).
Solution

This matches \(2\sin\theta\cos\theta=\sin 2\theta\) with \(\theta=3x\).

\(2\sin 3x\cos 3x\)\(=\)\(\sin(2\cdot 3x)\)
\(=\)\(\sin 6x\)
2 sine 3x cosine 3x equals sine 6x

Common pitfalls

\(\sin 2\theta\neq 2\sin\theta\). Use \(2\sin\theta\cos\theta\).
Half-angle sign from \(\theta/2\). Determine the quadrant of the half angle to pick \(+\) or \(-\).
Choose the right \(\cos 2\theta\) form. Using the wrong one forces an unnecessary Pythagorean step.

Frequently asked questions

What is the double-angle formula for sine?

\(\sin 2\theta=2\sin\theta\cos\theta\). It comes from the sine sum formula with both angles equal to \(\theta\).

Why does cos 2 theta have three forms?

Starting from \(\cos^2\theta-\sin^2\theta\), the Pythagorean identity lets you swap to \(1-2\sin^2\theta\) or \(2\cos^2\theta-1\).

How do you choose the sign in a half-angle formula?

From the quadrant of the half-angle \(\theta/2\): positive if the function is positive there, negative otherwise.

What are half-angle formulas used for?

Finding exact values of angles like \(15^\circ\) or \(22.5^\circ\), and rewriting powers of sine and cosine for integration later.

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