Trigonometric equations are vital for understanding numerous mathematical and scientific concepts. This guide is tailored for students, providing comprehensive strategies and solutions for solving trigonometric equations.

## Introduction to Trigonometric Equations

These equations involve trigonometric functions (sine, cosine, tangent, and their reciprocals) set equal to a value. The goal is to find all angles (usually represented as $t$ or $\theta$) that satisfy the equation.

### Strategies for Solving Trigonometric Equations

**1. Identifying the Function: **Ascertain which of the six primary trigonometric functions are involved.

**2. Isolating the Trigonometric Function: **Employ algebraic manipulation to isolate the function on one side of the equation.

**3. Using Inverse Trigonometric Functions:** Apply inverse functions to solve for the angle.

**4. Considering All Possible Angles:** Be aware of the periodic nature of these functions.

**5. Utilising Trigonometric Identities:** Use identities to simplify and solve equations.

### Methods to Isolate and Solve for Unknown Angles

**Algebraic Manipulation: **Techniques include expanding, factoring, or simplifying expressions.

**Graphical Interpretation:** Understanding the periodicity and symmetry of trigonometric functions through graphs.

**Substitution:** Use identities like $\sin^2(t) + \cos^2(t) = 1$ for simplification.

## Example Problems

### Example 1:

Solve $\tan(t) + \cot(t) = 4$:

**Solution:**

1. Rewrite $\cot(t)$ as $\frac{1}{\tan(t)}$: $\tan(t) + \frac{1}{\tan(t)} = 4$.

2. Clear the fraction by multiplying by:

3. Form a quadratic: $\tan^2(t) - 4\tan(t) + 1 = 0$.

4. Use quadratic formula: solutions are $t = \frac{\pi}{12}, \frac{5\pi}{12}$.

5. Account for periodicity: general solutions are $t = \frac{\pi}{12} + \pi k$ and $t = \frac{5\pi}{12} + \pi k$, where $k$ is an integer.

### Example 2:

Solve $2\sec(t) - 5\tan(t) = 2$:

**Solution:**

1. Use $\sec(t) = \frac{1}{\cos(t)}$: $\frac{2}{\cos(t)} - 5\tan(t) = 2$.

2. Multiply by$\cos(t)$: $2 - 5\sin(t) = 2\cos(t)$.

3. Rearrange: $2\cos(t) + 5\sin(t) = 2$.

4. Square and simplify using $\sin^2(t) + \cos^2(t) = 1$.

5. Identify that the equation holds for all $t$, with specific solutions at $t = 2\pi k$ and $t = \pi - \arctan\left(\frac{20}{21}\right) + 2\pi k$, where $k$ is an integer.

### Example 3:

Solve $3\cos^2(t) + \sin(t) = 1$:

**Solution:**

1. Use identity $\cos^2(t) = 1 - \sin^2(t) ): 3(1 - \sin^2(t)) + \sin(t) = 1$.

2. Form quadratic in $\sin(t) ): 3\sin^2(t) - \sin(t) - 2 = 0$.

3. Solve for $\sin(t)$ using quadratic formula.

4. Find $t$ using $\sin^{-1}$.

5. Account for periodicity: general solutions are $t = \frac{\pi}{2} + 2\pi k$, $t = -\arctan\left(\frac{2}{\sqrt{5}}\right) + 2\pi k$, and $t = -\pi + \arctan\left(\frac{2}{\sqrt{5}}\right) + 2\pi k$, where $k$ is an 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.