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Grade 12 · unit review

Review: Precalculus: Trigonometric Identities and Equations

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.

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Unit review questions
1. If sin θ=7/8, find cos²θ exactly.
Need a hint?
  • Unit-circle coordinates satisfy x²+y²=1, giving cos²θ+sin²θ=1.
  • Do not take a signed cosine value from its square alone.
2. If sin θ=-9/41 and cos θ=40/41, find sin(2θ).
Need a hint?
  • Sine of twice an angle equals twice the product of its sine and cosine.
  • Sine of twice an angle generally differs from twice its sine.
3. Find all solutions to sin θ=1/2 on 10π≤θ<12π.
Need a hint?
  • A positive sine level can occur twice in a full cycle, once on each side of its maximum.
  • The specified interval limits which repeated solutions belong in the answer.
4. Two triangle sides are 7 and 8 cm with included angle 120°. Use cos(120°)=-1/2 to find the exact opposite side.
Need a hint?
  • The law of cosines generalizes the right-triangle relationship by subtracting twice the side product times the included-angle cosine.
  • Match the included angle to the opposite requested side.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. If sin θ=9/10, find cos²θ exactly. Candidate answers: (A) 1/10 (B) 181/100 (C) 19/100 (D) 81/100. 2. 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. 3. Find all solutions to sin θ=1/2 on 14π≤θ<16π. Candidate answers: (A) 85/3π and 86/3π (B) 85/6π and 89/6π (C) 85/6π and 91/6π (D) 85/6π only. 4. Two triangle sides are 9 and 10 cm with included angle 90°. Use cos(90°)=0 to find the exact opposite side. Candidate answers: (A) 19 cm (B) √1 cm (C) √181 cm (D) √362 cm.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 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. 2. -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. 3. 85/6π and 89/6π. Within one cycle, sine 1/2 occurs at π/6 and 5π/6. Add the cycle shift 14π to both. The half-open interval contains both and avoids repeating an endpoint. 4. √181 cm. Law of cosines gives c²=9²+10²−2(9)(10)(0)=181. Therefore c=√181 cm. The included angle is opposite the requested side.

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