A horizontal line usually crosses one cycle of a sine curve twice, so most equations have two solutions in every 360°.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
Take the inverse to get the reference angle, decide which two quadrants fit the sign, then build both answers.
For sine
If sine is positive the answers sit in quadrants I and II, so they are θ and 180° − θ.
For cosine
If cosine is positive the answers sit in quadrants I and IV, so they are θ and 360° − θ.
Beyond one turn
Adding 360° to any solution gives another. If the interval is wider than one turn, keep adding until you leave it.
The circle gives the rest
After the calculator returns one solution, use the reference angle to find the second solution in the other quadrant where the function has that sign, then add multiples of the period for the infinitely many others.
Read the interval asked for
A question restricted to 0 ≤ x < 2π wants only the solutions in that window. Listing solutions outside it, or missing one inside it, are equally wrong. Sketching the interval on a circle prevents both.
The general solution
For sine, x = θ + 2πn or x = π − θ + 2πn. For cosine, x = ±θ + 2πn. For tangent, x = θ + πn, since its period is π. The n ranges over the integers and represents going round again.
Treat the argument as one object
For sin(2x) = 0.5, solve for 2x first over the doubled interval, then divide every solution by 2. Dividing too early loses half the solutions, which is the standard error in equations with a coefficient inside.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0.84)
Step 3: Watch an Example
One step at a time.
Watch Omar Solve sin θ = 0.5
Omar needs every solution between 0° and 360°.
- Step 1
The inverse sine of 0.5 is 30°, which is the reference angle.
Step 4: Your Turn
Practice makes it stick.
The Second Answer
Problem 1 of 2
sin θ = 0.643 gives 40° on a calculator. What is the other solution below 360°, in degrees?
degrees
The Ferris Wheel
Problem 2 of 2
A cabin height is h = 10 sin(30t) + 12 metres, t in minutes. At what value of 30t, in degrees, is the cabin first at 22 m?
degrees
Find Them All
1 of 8
sin θ = 0.5. The larger solution below 360°, in degrees?
2 of 8
cos θ = 0.5, one answer is 60°. The other below 360°?
3 of 8
sin θ = −0.5, reference 30°. The quadrant III solution, in degrees?
4 of 8
sin θ = −0.5. The quadrant IV solution, in degrees?
5 of 8
sin θ = 1. How many solutions between 0° and 360°?
6 of 8
sin θ = 2. How many solutions?
7 of 8
cos θ is negative. Which two quadrants hold the solutions?
8 of 8
sin θ = 0.5 with θ between 0° and 720°. How many solutions?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin θ = 0.866 gives 60°. 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.