In this section, we delve into the intricate world of solving trigonometric equations, a pivotal component of A-Level Pure Mathematics. Mastery of these techniques is essential for excelling in various mathematical challenges encountered in the curriculum.

Introduction

Solving trigonometric equations effectively hinges on understanding and applying various mathematical strategies. Here's a streamlined approach:

1. Isolate the Trigonometric Function: Move the trigonometric term (e.g., $sin$, $cos$, $tan$) to one side to simplify the equation.

2. Use Trigonometric Identities: Apply identities like the Pythagorean identity and angle sum/difference identities to simplify the equation into a more solvable form.

3. Acknowledge Periodicity: Understand that trigonometric functions repeat values at regular intervals, which is key to identifying all solution possibilities.

4. Apply Inverse Functions: Utilize inverse trigonometric functions to find the principal solution, serving as a starting point for all possible solutions.

5. Identify All Solutions: Considering the trigonometric function's periodicity, find all solutions within a given interval, ensuring a comprehensive solution set.

Examples

Example 1: $3\sin x + 1 = 0$

Solution:

1. Isolate $\sin x$:

$3\sin x = -1$

$\sin x = -\frac{1}{3}$

2. Principal Value:

$x = \arcsin\left(-\frac{1}{3}\right)$

The principal value is approximately $-0.34$ radians.

3. General Solutions:

Given the periodic nature of the sine function:

• First form: $x = \arcsin\left(-\frac{1}{3}\right) + 2n\pi$
• Second form: $x = \pi - \arcsin\left(-\frac{1}{3}\right) + 2n\pi$

Where $n$ is any integer, accounting for the sine function's periodicity of $2\pi$.

Example 2: $3\sin x - 5\cos x = 1$

Solution:

1. Trigonometric Identities Substitution:

$\sin x = \frac{2\tan\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}$,

$\cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}$

2. Substitute and Simplify:

Substitute these into $3\sin x - 5\cos x = 1$, resulting in a simplified equation in terms of $\tan\left(\frac{x}{2}\right)$.

3. Solve for $\tan\left(\frac{x}{2}\right)$:

Solutions for $\tan\left(\frac{x}{2}\right)$ are $-\frac{3}{4} + \frac{\sqrt{33}}{4}$ and $-\frac{3}{4} - \frac{\sqrt{33}}{4}$.

4. Find $x$ Using Inverse Tangent:

Calculate $x$ values: $x = 2\arctan\left(-\frac{3}{4} \pm \frac{\sqrt{33}}{4}\right)$

5. Periodicity and General Solutions:

Considering the periodicity of $\tan\left(\frac{x}{2}\right)$, which leads to a periodicity of $2\pi$ for $x$, the general solutions for $x$ include adding multiples of $2\pi$ to each solution.

General Solutions:

• $x = -2\arctan\left(\frac{3}{4} - \frac{\sqrt{33}}{4}\right) + 2n\pi$ and
• $x = -2\arctan\left(\frac{3}{4} + \frac{\sqrt{33}}{4}\right) + 2n\pi$, for $n$ being any integer.