Key takeaway: The double-angle formulas write the sine, cosine and tangent of 2θ using only trig functions of θ, so a doubled angle never needs a calculator.
Grade 12 – PrecalculusF-TF.C.9Trigonometry
Definition of Double-Angle Formulas
The double-angle formulas answer a simple question: if you know the sine and cosine of an angle, what are the sine and cosine of twice that angle? Doubling an angle does not double the trig value, so a rule is needed.
They follow from the sum formulas by setting both angles equal. Cosine of a doubled angle comes in three equivalent forms, and you pick whichever matches the information you have. They are handy for exact values, for rewriting squared sines and cosines, and for solving equations that mix \(\theta\) and \(2\theta\).
Formula and notation
The double-angle formulas for sine and cosine, valid for every angle \(\theta\):
\[\sin 2\theta=2\sin\theta\cos\theta,\qquad \cos 2\theta=2\cos^2\theta-1.\]
Two other forms of the cosine rule exist, one with \(\cos^2\theta-\sin^2\theta\) and one with \(1-2\sin^2\theta\). The tangent rule is twice \(\tan\theta\) divided by \(1-\tan^2\theta\).
Key facts
- The three cosine forms are interchangeable thanks to the Pythagorean identity.
- Solving the cosine forms for a squared sine or cosine gives the power-reducing rules.
- The tangent version fails when tan θ = ±1, because the denominator is zero.
- Plotting sin 2θ shows twice as many cycles as sin θ over the same interval.
Double-Angle Formulas example
Angle \(\theta\) lies in quadrant IV with \(\cos\theta=\frac{7}{25}\). Find \(\sin 2\theta\) and \(\cos 2\theta\).
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Find sine: \(\sin^2\theta=1-\frac{49}{625}=\frac{576}{625}\), and in quadrant IV it is negative, so \(\sin\theta=-\frac{24}{25}\).
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Sine of the doubled angle: twice the product of \(-\frac{24}{25}\) and \(\frac{7}{25}\), which is \(-\frac{336}{625}\).
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Cosine of the doubled angle: twice \(\frac{49}{625}\) minus 1, giving \(-\frac{527}{625}\).
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Check: \(336^2+527^2=112{,}896+277{,}729=390{,}625=625^2\).
\(\sin 2\theta=-\frac{336}{625}\) and \(\cos 2\theta=\)\(-\frac{527}{625}\).
Common mistake
Watch out
Joaquin claims \(\sin 2\theta=2\sin\theta\), so with \(\sin\theta=-\frac{24}{25}\) he gets \(-\frac{48}{25}\).
Why it is wrong
None of the double-angle formulas works that way. A sine can never reach \(-\frac{48}{25}\), since sine stays between \(-1\) and 1. The correct rule includes a cosine factor: \(\sin 2\theta=2\sin\theta\cos\theta=-\frac{336}{625}\).
Related terms and lessons
- Sum and Difference Formulas
- Trigonometric Identities
- Pythagorean Identity
- Unit Circle
- Period of a Function
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