A horizontal line usually crosses one cycle of a sine curve twice, so most equations have two solutions per turn.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
Take the inverse for the reference angle, decide which two quadrants match the sign, then build both answers.
For sine
If sine is positive the solutions are θ and 180° − θ. If negative, they sit in quadrants III and IV.
For cosine
If cosine is positive the solutions are θ and 360° − θ.
Beyond one turn
Adding 360° to any solution gives another. Keep adding until you leave the stated interval.
A doubled angle doubles the count
An equation in 2θ over one turn of θ means solving over two turns of 2θ, giving four solutions rather than two.
The calculator gives one of infinitely many
After the principal value, use the reference angle to find the second solution in the quadrant with the same sign, then add multiples of the period for all the rest.
Read the interval requested
A question restricted to 0 ≤ x < 2π wants exactly the solutions inside it. Listing extras or missing one are equally wrong, and sketching the circle prevents both.
A coefficient inside doubles the solutions
For sin(2x) = 0.5, solve for 2x over the doubled interval first, then divide every solution by 2. Dividing too early loses half the answers.
Identities reduce to one function
An equation mixing sine and cosine usually needs an identity to rewrite it in a single function first. Attempting to solve the mixed form directly rarely works.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0)
Step 3: Watch an Example
One step at a time.
Watch Diego Find Both Solutions
Diego solves sin θ = 0.5 for θ between 0° and 360°.
- Step 1
The inverse sine gives 30°, which is the reference angle.
Step 4: Your Turn
Practice makes it stick.
The Other Answer
Problem 1 of 2
sin θ = 0.5 gives 30°. What is the other solution below 360°, in degrees?
degrees
The Count
Problem 2 of 2
sin θ = 0.5 with θ between 0° and 720°. How many solutions?
Find Them All
1 of 8
cos θ = 0.5, one answer is 60°. The other below 360°?
2 of 8
sin θ = −0.5, reference 30°. The quadrant III solution, in degrees?
3 of 8
Same equation. The quadrant IV solution?
4 of 8
sin θ = 1. How many solutions between 0° and 360°?
5 of 8
sin θ = 2. How many solutions?
6 of 8
sin θ = 0.866 gives 60°. The other below 360°?
7 of 8
cos θ is negative. Which two quadrants hold the solutions?
8 of 8
cos θ = 1. What is θ in degrees, below 360°?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin θ = 0.643 gives 40°. The other solution below 360°, in degrees?
Question 2 of 2
Why does a calculator return only one solution?
What You Learned
- Most trigonometric equations have two solutions per 360°.
- Take the inverse for the reference angle, then place it in both correct quadrants.
- Add 360° repeatedly for solutions beyond one turn.