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.