Solving Trig Equations – Higher Mathematics SQA Revision

General Strategies for Solving Trig Equations

Identify the type of equation: Trig equations can involve just one trig function or a combination. Recognising the type of equation will guide the method of solution.
Use algebraic techniques: This could mean factorising, using the quadratic formula, or isolating a variable.
Use trig identities: Trig identities can help simplify and solve equations. For instance, you might use the Pythagorean identities to define one function in terms of another.
Consider the unit circle: Often, answers to trig equations correspond to angles in the unit circle. For the sine and cosine functions, this typically results in multiple or periodic solutions.

The solution to a basic trig equation like sin θ = 0.5 involves using inverse trig functions, denoted as sin^-1, cos^-1, or tan^-1.
For example: to solve sin θ = 0.5 for 0 ≤ θ < 2π, you might first use sin^-1(0.5) = θ to get one solution, then consider the symmetry of sine around its maximum and minimum points to find additional solutions.

These equations involve trig functions squared and therefore might look like quadratic equations.
To solve this type of equation, such as sin² θ - sin θ - 2 = 0 for 0 ≤ θ < 2π, re-write the equation in a quadratic form like (sin θ - 2)(sin θ + 1) = 0, and then solve for sin θ.

Trig functions are periodic, meaning they repeat their values in regular intervals. This often generates multiple solutions for trig equations within a specified domain.
For sin and cos, the period is 2π, while for tan it’s π.
If one solution is found, other solutions can be obtained by adding the period. For example, if θ = π/6 is a solution to sin θ = 0.5, then another solution would be θ = π/6 + 2πn, where n is an integer.

Always check your answers by plugging them back into the original equation to make sure they hold true.
Due to the nature of periodic functions, double-check that all derived solutions lie within the specified domain.