How to Solve Double Angle Identities?
A double angle formula is a trigonometric identity that expresses the trigonometric function \(2θ\) in terms of trigonometric functions \(θ\). In this step-by-step guide, you will learn more about double-angle formulas.
Solve Double Angle Identities: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Solve Double Angle Identities: pop-up practice
The double angle formulas are used to find the values of double angles of trigonometric functions using their single angle values. Also, the double-angle formulas can be used to derive the triple-angle formulas.
Related Topics
A step-by-step guide to double angle formulas
The double angle formulas are the special cases of the sum formulas of trigonometry and some alternative formulas are derived by using the Pythagorean identities. The sum formulas of trigonometry are:
- \(\color{blue}{sin\:\left(A\:+B\right)=sin\:A\:cos\:B\:+\:cos\:A\:sin\:B}\)
- \(\color{blue}{cos\:\left(A\:+\:B\right)=\:cos\:A\:cos\:B\:-\:sin\:A\:sin\:B}\)
- \(\color{blue}{tan\:\left(A\:+\:B\right)=\:\frac{\left(tan\:A\:+\:tan\:B\right)}{\left(1\:-\:tan\:A\:tan\:B\right)}}\)
What are double-angle formulas?
We derive double-angle formulas of \(sin, cos,\) and \(tan\) by substituting \(A=B\) in each of the above-sum formulas. Also, we will extract some alternative formulas that are derived using Pythagorean identities.
Double Angle Formulas , Example 1:
If \(tan A= \frac{3}{5}\), find the values of \(sin\:2A\).
Solution:
Since the value of \(tan\:A\) is given, we use the double angle formulas for finding \(sin\:2A\).
\(sin\:2A=\frac{2\:tan\:A}{1+tan^2A}\)
\(=\frac{2\left(\frac{3}{5}\right)^2}{1+\left(\frac{3}{5}\right)^2}\)
\(=\frac{\frac{18}{25}}{\frac{34}{25}}\)
\(=\frac{18\times 25}{25\times 34}\)
\(=\frac{9}{17}\)
Exercises for Double Angle Formulas
- Find a formula for \(cos(4x)\) in terms of \(cos x\).
- Solve the equation \(sin\:2x\:=\:cos\:x,\:0\:\le \:x\:<\pi\).
- \(\color{blue}{8\:cos^4x-8\:cos^2x+1}\)
- \(\color{blue}{x=\frac{\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}}\)
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