Recover a Squared Trigonometric Value

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: If sin θ=2/3, find cos²θ exactly. Candidate answers: (A) 1/3 (B) 13/9 (C) 4/9 (D) 5/9.
• Independent check inputs: If sin θ=3/4, find cos²θ exactly. Candidate answers: (A) 1/4 (B) 25/16 (C) 7/16 (D) 9/16.

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. Unit-circle coordinates satisfy x²+y²=1, giving cos²θ+sin²θ=1. A known sine determines cosine squared, but not cosine’s sign without quadrant information.

Here is the method on the supplied teaching case. The identity sin²θ+cos²θ=1 gives cos²θ=1−(2/3)²=5/9. It determines the square, while the sign of cos θ would still require quadrant information.

Do not take a signed cosine value from its square alone. The identity uses squared values, so subtract sin²θ rather than sinθ; the supplied result is an exact rational square quantity.

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=9/10, find cos²θ exactly. Candidate answers: (A) 1/10 (B) 181/100 (C) 19/100 (D) 81/100. Show the mathematical reason, not only the selected answer. Expected result: 19/100. The identity sin²θ+cos²θ=1 gives cos²θ=1−(9/10)²=19/100. It determines the square, while the sign of cos θ would still require quadrant information.

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