Solving Trigonometric Equations

The calculator gives one answer. The circle gives the rest.

12 problems · about 22 minutes · F-TF.B.7

What this lesson teaches

The student finds every solution of a basic trigonometric equation within a stated interval.

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.

Straightforward practice. Get the method working first.

5 problems

sin θ = 0.866 gives 60°. The other solution below 360°, in degrees?
Answer 120
Why 120°.
Why does a calculator return only one solution?
Answer The inverse function has a restricted range, so it must pick one.
Why The restricted range forces a single output.
sin θ = 0.5. The larger solution below 360°, in degrees?
Answer 150
Why 180 − 30.
cos θ = 0.5, one answer is 60°. The other below 360°?
Answer 300
Why 360 − 60.
sin θ = −0.5, reference 30°. The quadrant III solution, in degrees?
Answer 210
Why 180 + 30.

The same ideas with more to keep track of.

3 problems

sin θ = −0.5. The quadrant IV solution, in degrees?
Answer 330
Why 360 − 30.
sin θ = 1. How many solutions between 0° and 360°?
Answer 1
Why The peak is touched once.
sin θ = 2. How many solutions?
Answer 0
Why Sine never leaves −1 to 1.

Mixed problems. Work out what kind of question it is before you start.

4 problems

cos θ is negative. Which two quadrants hold the solutions?
Answer Quadrant II; Quadrant III
Why Cosine follows x, which is negative on the left half.
sin θ = 0.5 with θ between 0° and 720°. How many solutions?
Answer 4
Why Two per turn, and this is two turns.
The Second Answer: sin θ = 0.643 gives 40° on a calculator. What is the other solution below 360°, in degrees?
Answer 140 degrees
Why 140°.
The Ferris Wheel: 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?
Answer 90 degrees
Why 90°, the top of the wheel.