7.5 Trigonometric Equations

7.5.1 Learning Objectives

By the end of this section, you should be able to:

solve basic trigonometric equations;
give solutions on a specified interval;
write general solutions;
solve equations by factoring and identities; and
use inverse functions and Desmos for approximate solutions.

7.5.2 Basic Trigonometric Equations

First isolate the trigonometric function. Use the unit circle to find every angle in the required interval. For general solutions, include periodicity.

Example 7.13 Solving on One Cycle

Solve \(2\sin x-1=0\) on \([0,2\pi)\).

Solution

\[ \sin x=\frac12. \]

Sine is positive in Quadrants I and II:

\[ \boxed{x=\pi/6,\ 5\pi/6}. \]

Exercise 7.13 Solve \(\cos x=-\sqrt2/2\) on \([0,2\pi)\).

7.5.3 General Solutions

For sine and cosine, add \(2\pi k\) to each solution from one cycle. For tangent, add \(\pi k\):

\[ k\in\mathbb Z. \]

Example 7.14 Writing a General Solution

Solve \(\tan x=\sqrt3\) for all real \(x\).

Solution

The reference angle is \(\pi/3\). Since tangent has period \(\pi\),

\[ \boxed{x=\pi/3+k\pi,\quad k\in\mathbb Z}. \]

Exercise 7.14 Solve \(\cos x=0\) for all real \(x\).

7.5.4 Factoring and Identities

Move all terms to one side, factor, and apply the zero-product property. Identities may be needed to rewrite the equation in one trigonometric function.

Example 7.15 Solving by Factoring

Solve \(2\sin^2x-\sin x=0\) on \([0,2\pi)\).

Solution

\[ \sin x(2\sin x-1)=0. \]

Thus, \(\sin x=0\) or \(\sin x=1/2\):

\[ \boxed{x=0,\ \pi/6,\ 5\pi/6,\ \pi}. \]

Exercise 7.15 Solve \(2\cos^2x-1=0\) on \([0,2\pi)\).

7.5.5 Multiple-Angle and Approximate Equations

For equations involving \(Bx\), solve for the inner angle first, include all solutions that produce \(x\) in the required interval, and then divide by \(B\). For nonstandard values, use inverse functions or Desmos.

Example 7.16 Solving a Multiple-Angle Equation

Solve \(\sin2x=\sqrt3/2\) on \([0,2\pi)\).

Solution

Since \(0\le2x<4\pi\),

\[ 2x=\pi/3,\ 2\pi/3,\ 7\pi/3,\ 8\pi/3. \]

Divide by \(2\):

\[ \boxed{x=\pi/6,\ \pi/3,\ 7\pi/6,\ 4\pi/3}. \]

Exercise 7.16 Use Desmos to solve \(\sin x=0.3\) on \([0,2\pi)\) to three decimal places.

7.5.6 Conceptual Takeaways

Isolating the trigonometric function reveals the target unit-circle value.
Periodicity creates infinitely many general solutions.
Factoring can introduce several equation branches.
Multiple-angle equations require an expanded interval for the inner angle.
Every reported solution should satisfy both the original equation and requested interval.

7.5.7 Skills You Should Be Able to Do

Solve basic sine, cosine, and tangent equations.
List all solutions on a finite interval.
Write general solutions.
Solve equations using factoring and identities.
Find approximate solutions with inverse functions or Desmos.

7.5.8 Practice Problems with Solutions

Solve \(\sin x=-1\) on \([0,2\pi)\).

Show Solution

Sine is \(-1\) at the bottom of the unit circle:

\[ \boxed{x=3\pi/2}. \]
Solve \(2\cos x+1=0\) on \([0,2\pi)\).

Show Solution

\[ \cos x=-1/2. \]

Therefore, \(\boxed{x=2\pi/3,\ 4\pi/3}\).
Solve \(\tan x=-1\) for all real \(x\).

Show Solution

The principal solution is \(-\pi/4\), and tangent has period \(\pi\):

\[ \boxed{x=-\pi/4+k\pi,\quad k\in\mathbb Z}. \]
Solve \(\sin^2x-\sin x=0\) on \([0,2\pi)\).

Show Solution

\[ \sin x(\sin x-1)=0. \]

Thus, \(\sin x=0\) or \(\sin x=1\):

\[ \boxed{x=0,\ \pi/2,\ \pi}. \]
Solve \(2\cos^2x+\cos x-1=0\) on \([0,2\pi)\).

Show Solution

\[ (2\cos x-1)(\cos x+1)=0. \]

Therefore, \(\cos x=1/2\) or \(\cos x=-1\):

\[ \boxed{x=\pi/3,\ \pi,\ 5\pi/3}. \]
Solve \(\cos3x=0\) on \([0,2\pi)\).

Show Solution

Since \(\cos u=0\) when \(u=\pi/2+k\pi\),

\[ 3x=\frac{\pi}{2}+k\pi. \]

Thus,

\[ x=\frac{\pi}{6}+\frac{k\pi}{3}. \]

Taking \(k=0,1,2,3,4,5\) gives

\[ \boxed{x=\pi/6,\ \pi/2,\ 5\pi/6,\ 7\pi/6,\ 3\pi/2,\ 11\pi/6}. \]