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.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.