Ratios and Graphs of Trigonometric Functions
How to Solve a trigonometric equation
Use this to find all angles between 0° and 360° that satisfy a trigonometric equation.
Before you start
- Finding sin, cos and tan on a calculator
- Using the ASTC quadrant sign rule
- Rearranging a simple equation to isolate one term
When to use it
Use this to find all angles between 0° and 360° that satisfy a trigonometric equation.
The steps
- Rearrange to get the trig ratio alone, e.g. sin x = 0.5.
- Find the basic angle from the calculator, in degrees.
- Decide which quadrants give the required sign.
- Use the basic angle to write the angle in each valid quadrant.
- List all solutions in the range 0° to 360°.
Worked example
Solve 2 sin x = 1 for 0° ≤ x ≤ 360°.
- Rearrange to get the ratio alone: divide by 2 to get sin x = 0.5.
- Basic angle from the calculator: sin⁻¹(0.5) = 30°.
- sin x is positive, so x is in Q1 and Q2, where sine is positive.
- Write the angle in each quadrant: Q1 gives x = 30°; Q2 gives x = 180° − 30° = 150°.
- In the range 0° ≤ x ≤ 360° the solutions are x = 30° and x = 150°.
A second example, with a twist
The ratio comes out negative, so the answers fall in the second and third quadrants and the value needs rounding. Solve 3 cos x + 2 = 0 for 0° ≤ x ≤ 360°, giving x to the nearest degree.
- Rearrange to get the ratio alone: 3 cos x = −2, so cos x = −2/3 ≈ −0.6667.
- Basic angle from the positive value: cos⁻¹(0.6667) = 48° (to the nearest degree).
- cos x is negative, so x is in Q2 and Q3, where cosine is negative.
- Write the angle in each quadrant: Q2 gives x = 180° − 48° = 132°; Q3 gives x = 180° + 48° = 228°.
- In the range 0° ≤ x ≤ 360° the solutions are x = 132° and x = 228°.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Why do I get two answers instead of one?
Between 0° and 360° each ratio value is shared by two angles, one in each quadrant where the sign matches. The calculator gives only the basic angle, so you must add the second angle yourself using 180° − θ, 180° + θ or 360° − θ.
Some equations have even more solutions.
Do I use the negative sign when finding the basic angle?
No. Always take the inverse of the positive value to get the basic acute angle, then let the quadrants decide the sign.
If you type the negative into the calculator you may get an angle outside 0° to 360° and miss some solutions, so drop the minus sign first.
What if the range is bigger, like 0° to 720°?
Find all the solutions in one full turn first, then add 360° to each to reach the next turn, repeating until you pass the top of the range. For 0° to 720° you list each first-turn answer and each answer plus 360°, keeping only those that stay inside the range.