Strategy for Trigonometric Equations

Before solving, identify if the argument of the trigonometric function is a simple variable ($x$) or a compound function (e.g., $kx$ or $x \pm \alpha$). If the argument is modified, the search interval must be adjusted accordingly to ensure no solutions are missed.

For an equation like $\cos(2x) = k$ with a range $0^\circ \le x \le 180^\circ$, the working range must be transformed to $0^\circ \le 2x \le 360^\circ$. This ensures that all cycles of the function within the original bounds are accounted for during the calculation phase.

Equations containing multiple trigonometric functions (e.g., both $\sin x$ and $\cos x$) must be simplified into a single trigonometric ratio before they can be solved. This is typically achieved using fundamental identities.

The identity $\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$ is used to convert linear combinations of sine and cosine into a single tangent function. The Pythagorean identity $\sin^2 \theta + \cos^2 \theta \equiv 1$ is essential for quadratic equations, allowing the substitution of squared terms to create a consistent equation in one variable.

When an equation involves squared trigonometric terms, it often takes a quadratic form such as $a\sin^2 x + b\sin x + c = 0$. These are solved by treating the trigonometric function as a single variable (e.g., let $u = \sin x$) and factorising the resulting quadratic expression.

After factorising, solve for the trigonometric ratio (e.g., $\sin x = k$). It is critical to check if these values are valid; for sine and cosine, solutions only exist if $-1 \le k \le 1$. If a factor yields a value outside this range, that specific branch of the equation has no real solutions.

• Check Calculator Mode: Always verify if the question requires degrees or radians before starting. A common error is calculating in degrees when the interval is given in terms of $\pi$.
• The 'Add/Subtract Period' Rule: Once the primary solutions within one cycle ($0$ to $360^\circ$ or $2\pi$) are found, add or subtract the period (e.g., $360^\circ$) to find all other solutions in the specified range.
• Don't Divide by Trig Functions: Never divide both sides of an equation by a trigonometric function (like $\sin x$), as this can result in the loss of valid solutions where that function equals zero. Instead, factorise the expression.