Double-angle identities
Objective. Derive and select equivalent double-angle forms.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Double-angle identities are special cases of compound-angle formulae with A=B, giving sin2A=2sinAcosA, cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A, and tan2A=2tanA/(1−tan²A). Example: knowing sinA=3/5 and cosA=4/5, cos2A=2(4/5)²−1=7/25, avoiding the need to find angle A directly. Choosing the correct equivalent form of cos2A often simplifies an equation more than others, depending on whether sin or cos terms dominate.
Worked example
Given that sinA=5/13 and A is acute, find the exact value of cos2A using double-angle identities.
- 1Find cosA using sin²A+cos²A=1: cosA=√(1−25/169)=12/13
- 2Select the form cos2A=1−2sin²A
- 3Substitute sinA=5/13: cos2A=1−2(25/169)
- 4Simplify: 1−50/169=119/169
Answer: 119/169
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Start practisingCommon mistake. Students write sin2A=2sinA, omitting the necessary cosA factor; always include both sinA and cosA multiplied together in the double-angle sine identity.
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