Solving Trigonometric Equations 5.5

Trigonometric Equations

Introduction to Solving Trigonometric Equations

Solving trigonometric equations is a fundamental skill in precalculus and calculus. These equations involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. The main strategy is to isolate the trigonometric function, apply the appropriate inverse function, and use the unit circle to find all possible solutions within a given interval or as a general solution.

• Unit Circle: The unit circle provides exact values for sine and cosine at key angles between 0 and 2π radians (or 0° to 360°).

• Inverse Trigonometric Functions: Used to solve for the angle once the trigonometric function is isolated.

• General Solutions: If no interval is specified, solutions must include all coterminal angles by adding multiples of the period (e.g., 2πk for sine and cosine).

Solving Basic Trigonometric Equations

Isolating the Trigonometric Function

To solve equations such as $\cos x = \frac{2}{3}$ or $\sin x = -\frac{1}{2}$, first isolate the trigonometric function, then apply the inverse function to both sides:

• Example: Solve $\cos x = \frac{2}{3}$ for $x$ on $[0, 360^\circ)$.

• Take the inverse cosine: $x = \cos^{-1}\left(\frac{2}{3}\right)$.

• Find all angles on the unit circle where the x-coordinate is $\frac{2}{3}$: $x = 30^\circ, 330^\circ$.

General Solutions in Radians

If no interval is given, express the solution in terms of all possible coterminal angles:

• Example: Solve $\sin x = -\frac{1}{2}$.

• Isolate and invert: $x = \sin^{-1}\left(-\frac{1}{2}\right)$.

• Unit circle solutions: $x = \frac{7\pi}{6}, \frac{11\pi}{6}$.

• General solution: $x = \frac{7\pi}{6} + 2\pi k, \frac{11\pi}{6} + 2\pi k$ for integer $k$.

Solving Equations with Quadratic Forms

Using Square Roots and Factoring

Some equations require taking square roots or factoring before solving:

• Example: Solve $\cos^2 x = \frac{3}{4}$ on $[0, \frac{\pi}{2})$.

• Take square roots: $\cos x = \pm\frac{\sqrt{3}}{2}$.

• Find all angles in the interval with these cosine values: $x = \frac{\pi}{6}$ (since only positive values are in $[0, \frac{\pi}{2})$).

Solving Equations Involving Reciprocal Trig Functions

Converting to Sine or Cosine

Equations with cosecant, secant, or cotangent are often rewritten in terms of sine, cosine, or tangent:

• Example: Solve $5\csc x = -3$ on $(0, 2\pi)$.

• Isolate: $\csc x = -\frac{3}{5}$.

• Recall $\csc x = \frac{1}{\sin x}$, so $\sin x = -\frac{5}{3}$ (not possible since $|\sin x| \leq 1$). If instead $\csc x = 1$, then $\sin x = 1$.

• Unit circle solution: $x = \frac{\pi}{2}$.

Solving Equations with Multiple Angles or Arguments

General Solutions with Multiples

When the argument of the trig function is not just $x$ (e.g., $\sin(2x)$), solve for the argument first, then for $x$:

• Example: Solve $\sin(2\theta) = -\frac{\sqrt{3}}{2}$ on $(0, 2\pi)$.

• Find all $2\theta$ where $\sin(2\theta) = -\frac{\sqrt{3}}{2}$: $2\theta = \frac{4\pi}{3}, \frac{5\pi}{3}$.

• General solution: $2\theta = \frac{4\pi}{3} + 2\pi k, \frac{5\pi}{3} + 2\pi k$.

• Divide by 2: $\theta = \frac{2\pi}{3} + \pi k, \frac{5\pi}{6} + \pi k$.

Solving Trigonometric Equations Using the Zero Product Property

Factoring and Setting Each Factor to Zero

When the equation is a product of factors, set each factor to zero and solve separately:

• Example: Solve $(\tan x - 1)(\cos x + 1) = 0$ on $(0, 2\pi)$.

• Set each factor to zero:

  • $\tan x - 1 = 0 \implies \tan x = 1 \implies x = \frac{\pi}{4}, \frac{5\pi}{4}$

  • $\cos x + 1 = 0 \implies \cos x = -1 \implies x = \pi$

• All solutions: $x = \frac{\pi}{4}, \frac{5\pi}{4}, \pi$

Solving Equations with Trigonometric Identities

Using Identities to Simplify Equations

Some equations require using identities to rewrite all terms in terms of a single trigonometric function:

• Example: Solve $2\cos^2 \theta - 2\sin^2 \theta = 0$ on $(0, 2\pi)$.

• Use the identity $\cos 2\theta = 2\cos^2 \theta - 1$ or $\cos^2 \theta = 1 - \sin^2 \theta$ to rewrite the equation.

• Alternatively, factor or use Pythagorean identities as needed.

Summary Table: Common Steps in Solving Trigonometric Equations

Step	Description	Example
Isolate the trig function	Move all terms to get the trig function alone on one side	$2\sin x + 1 = 0 \implies \sin x = -\frac{1}{2}$
Apply inverse function	Take the appropriate inverse trig function of both sides	$x = \sin^{-1}\left(-\frac{1}{2}\right)$
Find all solutions in interval	Use the unit circle to find all angles in the given interval	$x = \frac{7\pi}{6}, \frac{11\pi}{6}$
Write general solution	If no interval, add multiples of the period	$x = \frac{7\pi}{6} + 2\pi k$
Use identities if needed	Rewrite using Pythagorean, double-angle, or reciprocal identities	$\sin^2 x + \cos^2 x = 1$
Factor and use Zero Product Property	Set each factor to zero and solve	$(\sin x)(2\cos x - 1) = 0$

Key Trigonometric Identities Used in Solving Equations

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

• Double Angle for Cosine: $\cos 2x = 2\cos^2 x - 1$

• Reciprocal Identities: $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{1}{\tan x}$

Examples and Applications

• Example 1: Solve $\cos x = 0$ on $(0, 2\pi)$ Solution: $x = \frac{\pi}{2}, \frac{3\pi}{2}$

• Example 2: Solve $\sin^2 x = \frac{1}{4}$ Solution: $\sin x = \pm\frac{1}{2} \implies x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$

• Example 3: Solve $\tan x = \sqrt{3}$ on $[0, 2\pi)$ Solution: $x = \frac{\pi}{3}, \frac{4\pi}{3}$

Additional info:

• When solving for all solutions, always add $2\pi k$ (for sine and cosine) or $\pi k$ (for tangent and cotangent) to account for periodicity, where $k$ is any integer.

• Check the domain of the trigonometric function; for example, $|\sin x| \leq 1$ and $|\cos x| \leq 1$.

• Always express answers in the units (degrees or radians) specified by the problem.