Double-Angle Formulas
In short: The double-angle formulas express sin 2θ, cos 2θ and tan 2θ using only the trig values of θ itself, for instance sin 2θ = 2 sin θ cos θ.
PrecalculusTrigonometryF-TF.C.9
What is Double-Angle Formulas?
The double-angle formulas answer a common puzzle: if you know the trig values of an angle, what are the values for twice that angle? You never have to find the angle itself; products and squares of the known values are enough.
They are \(\sin 2\theta = 2\sin\theta\cos\theta\); \(\cos 2\theta = \cos^2\theta – \sin^2\theta = 2\cos^2\theta – 1 = 1 – 2\sin^2\theta\); and \(\tan 2\theta = \frac{2\tan\theta}{1 – \tan^2\theta}\). The three forms of the cosine rule are equivalent, thanks to the Pythagorean identity.
How to spot it
- A trig value of 2θ is wanted, but only sin θ or cos θ is given, which is the job of the double-angle formulas.
- An expression has the shape 2 sin x cos x, or cos²x – sin²x, which hints at a hidden doubled angle.
- An equation mixes sin 2x with sin x or cos x and must be factored to be solved.
- An exact value for an angle like 15° or 75° is requested, using a half or double of a familiar angle.
The method
- Find the missing trig value of \(\theta\) first, using a triangle or the Pythagorean identity.
- Choose the form of the double-angle formulas that uses what you know: for cosine, any of the three versions.
- Substitute the values and multiply or square with care.
- Check the result: \(\sin^2 2\theta + \cos^2 2\theta\) should equal 1.
Double-Angle Formulas: solved example
Problem
An acute angle \(\theta\) has \(\sin\theta = \frac{8}{17}\) and \(\cos\theta = \frac{15}{17}\). Find \(\sin 2\theta\) and \(\cos 2\theta\).
Solution
Sine: \(\sin 2\theta = 2\cdot\frac{8}{17}\cdot\frac{15}{17} = \frac{240}{289}\). Cosine: \(\cos 2\theta = \frac{225}{289} – \frac{64}{289} = \frac{161}{289}\).
Check: \(240^2 + 161^2 = 57{,}600 + 25{,}921 = 83{,}521 = 289^2\). The answers are \(\sin 2\theta = \frac{240}{289}\) and \(\cos 2\theta = \frac{161}{289}\).
Try it
Question 1
If \(\cos\theta = \frac{1}{4}\), find \(\cos 2\theta\).
Show the answer
\(2\cos^2\theta – 1 = 2\cdot\frac{1}{16} – 1 = -\frac{7}{8}\).
Question 2
Evaluate \(\sin 15^{\circ}\cos 15^{\circ}\) exactly.
Show the answer
It equals \(\frac{1}{2}\sin 30^{\circ} = \frac{1}{4}\).
Double-Angle Formulas vs. Sum and Difference Formulas
Doubling is the special case in which both angles being added are the same, so \(\sin(\theta + \theta)\) becomes \(2\sin\theta\cos\theta\). The sum and difference rules handle two different angles, such as 45° and 30°. Remember that \(\sin 2\theta\) is not \(2\sin\theta\); the angle inside changes the value.
Keep going
Lessons
Worksheets and quizzes
Related words
- Sum and Difference Formulas
- Trigonometric Identities
- Pythagorean Identity
- Unit Circle
- Inverse Trigonometric Functions
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