Use the Double-Angle Sine Identity

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: If sin θ=3/5 and cos θ=4/5, find sin(2θ). Candidate answers: (A) 12/25 (B) 24/25 (C) 6/5 (D) 7/5.
• Independent check inputs: If sin θ=-5/13 and cos θ=12/13, find sin(2θ). Candidate answers: (A) -10/13 (B) -120/169 (C) -60/169 (D) 7/13.

Learn the idea

An identity holds for all inputs in its domain, while an equation asks for particular angle solutions. Algebraic identities, angle formulas and geometric laws must retain signs, intervals and their defining conditions. Sine of twice an angle equals twice the product of its sine and cosine. The formula links angle addition to supplied coordinate values and preserves both factors’ signs.

Here is the method on the supplied teaching case. The supplied sine and cosine squares total 1. Use sin(2θ)=2sinθcosθ=2(3/5)(4/5)=24/25; doubling the angle does not simply double its sine.

Sine of twice an angle generally differs from twice its sine. Use both supplied ratios and confirm they are compatible with a unit-circle point; dropping cosine can exceed the allowable sine range.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: If sin θ=-11/61 and cos θ=60/61, find sin(2θ). Candidate answers: (A) -1320/3721 (B) -22/61 (C) -660/3721 (D) 49/61. Show the mathematical reason, not only the selected answer. Expected result: -1320/3721. The supplied sine and cosine squares total 1. Use sin(2θ)=2sinθcosθ=2(-11/61)(60/61)=-1320/3721; doubling the angle does not simply double its sine.

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