Trigonometric Equations

Solve sine, cosine, and tangent equations with the unit circle and periodicity.

A trigonometric equation asks for angle values that make a trig statement true. Unlike many algebra equations, trig equations often have repeating answers. If one angle works, every coterminal angle may also work.

The main idea is simple:

algebra first, unit circle second, periodicity third

Learning objectives

After this lesson you should be able to:

  • solve basic equations such as sin(theta)=1/2, cos(theta)=0, and tan(theta)=1;
  • find all solutions in a requested interval;
  • write infinite solution families using periods;
  • isolate a trig function before using the unit circle;
  • check candidate solutions and reject extraneous ones;
  • use SymPy as a verification tool while still reasoning from the unit circle.

Why trig equations have many answers

On the unit circle, each angle gives a point (cos(theta), sin(theta)). Since a full turn brings you back to the same point, trig values repeat.

For example:

sin(30 degrees) = 1/2
sin(150 degrees) = 1/2
sin(390 degrees) = 1/2

The angles 30 degrees and 390 degrees are coterminal, and 150 degrees has the same reference angle but is in a different quadrant.

The solving routine

Use this routine whenever you see a trig equation.

  1. Read the interval. Are answers requested in degrees, radians, or all real angles?
  2. Isolate one trig expression. For example, change 2cos(theta)-1=0 into cos(theta)=1/2.
  3. Find the reference angle. Use special angles or inverse trig.
  4. Choose quadrants using the sign. Positive sine means Quadrants I or II; negative cosine means Quadrants II or III; positive tangent means Quadrants I or III.
  5. List answers in the interval. Do not stop after the first angle.
  6. For all real solutions, add the period. Sine and cosine repeat every 360 degrees or 2pi; tangent repeats every 180 degrees or pi.
  7. Check. Substitute each answer mentally, symbolically, or numerically.

Basic unit-circle equations

Some equations can be read directly from the unit circle.

sin(theta) = 0      -> theta is on the x-axis
cos(theta) = 0      -> theta is on the y-axis
tan(theta) = 0      -> sine is 0 and cosine is not 0

In [0,360):

sin(theta)=0  -> 0, 180 degrees
cos(theta)=0  -> 90, 270 degrees
tan(theta)=0  -> 0, 180 degrees

Reference angles and quadrants

If the equation uses a special value, first find the reference angle.

sin(theta)=1/2      -> reference angle 30 degrees
cos(theta)=sqrt(2)/2 -> reference angle 45 degrees
tan(theta)=sqrt(3) -> reference angle 60 degrees

Then use signs.

Equation Positive or negative? Quadrants
sin(theta)=positive positive I, II
sin(theta)=negative negative III, IV
cos(theta)=positive positive I, IV
cos(theta)=negative negative II, III
tan(theta)=positive positive I, III
tan(theta)=negative negative II, IV

Worked example 1: solve a sine equation in degrees

Solve in [0,360):

sin(theta)=1/2

Step 1: Identify the reference angle.

sin(30 degrees)=1/2

So the reference angle is 30 degrees.

Step 2: Decide the quadrants.

Sine is positive because the right side is positive. Sine is positive in Quadrants I and II.

Step 3: Build the angles.

Quadrant I angle:  30 degrees
Quadrant II angle: 180 - 30 = 150 degrees

Answer:

theta = 30 degrees or 150 degrees

Check. Both angles have y-coordinate 1/2 on the unit circle.

Worked example 2: isolate before solving

Solve in [0,2pi):

2cos(theta) - 1 = 0

Step 1: Isolate cosine.

2cos(theta) - 1 = 0
2cos(theta) = 1
cos(theta) = 1/2

Step 2: Find the reference angle.

cos(pi/3)=1/2

The reference angle is pi/3.

Step 3: Choose quadrants.

Cosine is positive in Quadrants I and IV.

Step 4: List answers in [0,2pi).

Quadrant I:  pi/3
Quadrant IV: 2pi - pi/3 = 5pi/3

Answer:

theta = pi/3 or 5pi/3

Worked example 3: write all real solutions

Solve for all real theta:

tan(theta)=1

Step 1: Find one angle.

tan(45 degrees)=1

In radians, that is pi/4.

Step 2: Use tangent’s period.

Tangent repeats every pi, not every 2pi.

Answer:

theta = pi/4 + k*pi, where k is any integer

This compact formula includes pi/4, 5pi/4, -3pi/4, and infinitely many more.

Worked example 4: factor first

Solve in [0,360):

2sin^2(theta) - sin(theta) = 0

Step 1: Factor.

2sin^2(theta) - sin(theta)
= sin(theta)(2sin(theta)-1)

So:

sin(theta)(2sin(theta)-1)=0

Step 2: Use the zero product property.

sin(theta)=0

or

2sin(theta)-1=0
sin(theta)=1/2

Step 3: Solve each equation.

For sin(theta)=0 in [0,360):

theta = 0, 180 degrees

For sin(theta)=1/2 in [0,360):

theta = 30, 150 degrees

Answer:

theta = 0, 30, 150, 180 degrees

Common pitfalls

  • Forgetting to isolate the trig function before reading the unit circle.
  • Giving only the reference angle and missing the second quadrant solution.
  • Using sine/cosine period 2pi for tangent; tangent’s period is pi.
  • Mixing degrees and radians in the same answer.
  • Dividing by a trig expression that could be zero and losing solutions.
  • Including 360 degrees in [0,360); the square bracket/parenthesis means 0 is included but 360 is not.

Practice exercises

  1. Solve in [0,360): cos(theta)=1/2.
  2. Solve in [0,360): sin(theta)=-1/2.
  3. Solve in [0,2pi): tan(theta)=0.
  4. Solve for all real x: sin(x)=0.
  5. Solve in [0,360): 2cos(theta)+1=0.
  6. Solve in [0,360): sin(theta)(sin(theta)-1)=0.
  1. cos(theta)=1/2 has reference angle 60 degrees; cosine is positive in Quadrants I and IV, so theta=60, 300 degrees.
  2. sin(theta)=-1/2 has reference angle 30 degrees; sine is negative in Quadrants III and IV, so theta=210, 330 degrees.
  3. Tangent is zero where sine is zero and cosine is not zero: x=0, pi.
  4. x=k*pi, where k is any integer.
  5. 2cos(theta)+1=0 gives cos(theta)=-1/2; reference angle 60 degrees, Quadrants II and III, so theta=120, 240 degrees.
  6. sin(theta)=0 or sin(theta)=1, so theta=0, 180, 90 degrees. In increasing order: 0, 90, 180 degrees.

Subtopic guided practice and checkpoints

These small labs separate the skills that are usually mixed together in a full problem.

Lab 1: isolate first

Guess first. What is the first algebra move in 3sin(theta)+2=5?

Guided exercise.

Subtract 2, then divide by 3:

3sin(theta)+2=5
3sin(theta)=3
sin(theta)=1

Now the unit circle can be used.

Checkpoint. Isolate the trig function in 4cos(theta)-2=0.

Lab 2: find the reference angle

Guess first. If sin(theta)=sqrt(2)/2, what special angle should you think of?

Guided exercise.

The value sqrt(2)/2 belongs to 45 degrees or pi/4.

Checkpoint. What is the reference angle for cos(theta)=sqrt(3)/2?

Lab 3: choose all quadrants

Guess first. If cosine is negative, which quadrants are possible?

Guided exercise.

Cosine is the x-coordinate. It is negative on the left side of the unit circle, so Quadrants II and III are possible.

Checkpoint. If tangent is negative, which quadrants are possible?

Lab 4: write a periodic family

Guess first. Does cos(x)=1 repeat every pi or 2pi?

Guided exercise.

Cosine repeats every 2pi, so all solutions are:

x = 2k*pi, where k is any integer

Checkpoint. Write all real solutions of tan(x)=0.

Lab 5: check the interval

Guess first. Should 360 degrees be included in [0,360)?

Guided exercise.

No. The notation [0,360) means include 0 and exclude 360. This avoids counting the same direction twice.

Checkpoint. In [0,2pi), should 2pi be listed?

  1. cos(theta)=1/2.
  2. 30 degrees or pi/6.
  3. Quadrants II and IV.
  4. x=k*pi, where k is any integer.
  5. No, 2pi is excluded from [0,2pi).

Trig equation guessing game

Using this lesson with edumath and SymPy

Use edumath for angle facts and SymPy for algebraic verification. SymPy can solve many equations, but interval reasoning is still easier when you understand the unit circle.

from edumath.trigonometry import reference_angle_degrees, trig_signs

reference_angle_degrees(150)
30
trig_signs(150)
{'sine': '+', 'cosine': '-', 'tangent': '-'}

Use SymPy to solve or verify exact equations.

import sympy as sp

theta = sp.symbols("theta")
sp.solve(sp.Eq(2 * sp.cos(theta) - 1, 0), theta)
[pi/3, 5*pi/3]

For all real solutions, SymPy’s solveset gives a set description.

sp.solveset(sp.Eq(sp.sin(theta), sp.Rational(1, 2)), theta, domain=sp.S.Reals)

\(\displaystyle \left\{2 n \pi + \frac{\pi}{6}\; \middle|\; n \in \mathbb{Z}\right\} \cup \left\{2 n \pi + \frac{5 \pi}{6}\; \middle|\; n \in \mathbb{Z}\right\}\)