Study guide: 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.
Ideas to explain in your own words
- 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. Do not take a signed cosine value from its square alone.
- 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. Sine of twice an angle generally differs from twice its sine.
- A positive sine level can occur twice in a full cycle, once on each side of its maximum. Reference-angle solutions repeat after adding whole multiples of two π. The specified interval limits which repeated solutions belong in the answer.
- The law of cosines generalizes the right-triangle relationship by subtracting twice the side product times the included-angle cosine. A sixty-degree included angle has exact cosine one half. Match the included angle to the opposite requested side.
Useful vocabulary
- identity
- An equation true for every input in its common domain.
- double-angle formula
- A trigonometric relationship expressing a function at twice an angle.
- law of cosines
- c²=a²+b²−2ab cosC, with C opposite c.
Watch your reasoning
- 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.
- 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.
- The specified interval limits which repeated solutions belong in the answer. Keep both within-cycle solutions and respect half-open endpoints; an inverse-sine calculator value alone usually gives only the principal branch.
A worked review example
Complete supplied guide examples: Recover a Squared Trigonometric Value — If sin θ=2/3, find cos²θ exactly. Candidate answers: (A) 1/3 (B) 13/9 (C) 4/9 (D) 5/9. 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. Use the Double-Angle Sine Identity — 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. 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. Find Every Solution in a Stated Cycle — Find all solutions to sin θ=1/2 on 0π≤θ<2π. Candidate answers: (A) 1/3π and 2/3π (B) 1/6π and 5/6π (C) 1/6π and 7/6π (D) 1/6π only. Within one cycle, sine 1/2 occurs at π/6 and 5π/6. Add the cycle shift 0π to both. The half-open interval contains both and avoids repeating an endpoint. Use an Included Angle in a Nonright Triangle — Two triangle sides are 2 and 3 cm with included angle 60°. Use cos(60°)=1/2 to find the exact opposite side. Candidate answers: (A) 5 cm (B) √1 cm (C) √26 cm (D) √7 cm. Law of cosines gives c²=2²+3²−2(2)(3)(1/2)=7. Therefore c=√7 cm. The included angle is opposite the requested side.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.