Trigonometric Equations with Identities

The Role of Identities in Solving Equations

A trigonometric equation becomes much harder to solve when it involves more than one distinct trigonometric function or when it contains a mix of single and double angles. The key insight is to use identities to reduce the equation to a single function of a single angle — after which standard algebraic methods apply. This is the same principle used when solving any equation: simplify until it becomes recognisable.

Converting to a Single Ratio Using Pythagorean Identities

When an equation contains both sinθ and cosθ but no double angle terms, look for the Pythagorean identity sin²θ + cos²θ = 1 to eliminate one of the ratios. For example, if you see sin²θ in an equation that also contains cosθ, replace sin²θ with 1 − cos²θ to create a quadratic in cosθ. Similarly, replace cos²θ with 1 − sin²θ when sinθ is the simpler remaining function. Choose the substitution that reduces the equation to a single variable.

Factorising After Substitution

After applying an identity, you will typically obtain a quadratic (or occasionally higher-degree polynomial) in sinθ or cosθ. Treat this exactly as an algebraic quadratic: let u = sinθ (or cosθ), factor or use the quadratic formula, then solve for u. Remember to check that your solutions for u are within the range of the trig function — sinθ and cosθ must lie in [−1, 1], and any value outside this range gives no solution. Always write out all cases.

Double Angle Equations

When the equation involves sin 2θ or cos 2θ, replace these using the double angle formulas and express everything in terms of single-angle functions. For sin 2θ = sinθ, substitute 2 sinθ cosθ and rearrange to get sinθ(2cosθ − 1) = 0. This factorised form immediately gives the cases. For cos 2θ equations, choose the form of cos 2θ that matches the other terms — if the equation also contains cosθ, use cos 2θ = 2cos²θ − 1; if it contains sinθ, use cos 2θ = 1 − 2sin²θ.

Finding All Solutions in a Given Interval

Once you have found the reference angle α (the acute angle solving |sinθ| = k or |cosθ| = k), use the ASTC rule to determine which quadrants give valid solutions, then list all angles in the specified interval. For radians: for sinθ = k, the two solutions in [0, 2π) are α and π − α when k > 0, or π − α and 2π + α when k < 0 (equivalently α and π − α adjusted). Always verify each solution by substituting back into the original equation.

General Solutions

When the question asks for the general solution rather than solutions in a specific interval, express the answer using an integer parameter n. For example, sinθ = 1/2 gives the general solution θ = π/6 + 2πn or θ = 5π/6 + 2πn, n ∈ ℤ. This concisely captures all solutions across the entire real line. Tan equations have a period of π, so tanθ = k gives θ = arctan(k) + πn.

Common Pitfalls

The most frequent error is dividing by a trigonometric expression (like sinθ or cosθ) instead of factorising. Division loses solutions where the divisor is zero. A second common error is selecting the wrong form of cos 2θ: using cos 2θ = 2cos²θ − 1 when the equation contains sinθ creates a mixed equation rather than simplifying it. Always choose the form that eliminates the second function. Finally, always check solutions in the original equation in case a step (such as squaring) introduced extraneous solutions.