A trig equation usually has more than one solution, because the graphs repeat. For sin x = 0.2 in 0° ≤ x < 360°, the calculator gives 11.5°. The sine graph is symmetric about 90°, so the other solution is 180° − 11.5° = 168.5°.
For cos x = k, the second solution is 360° minus the first. For tan x = k, add 180°. Use a sketch of the graph or the CAST diagram to see which quadrants contain solutions, and check that each answer is inside the interval.
Worked example
Solve cos x = 0.5 for 0° ≤ x < 360°.
- cos⁻¹ 0.5 = 60°.
- Second solution: 360° − 60° = 300°.
Answer:x = 60° or x = 300°
Key words
- principal value
- the answer the calculator gives
- interval
- the range of x allowed
- CAST diagram
- shows where sin, cos and tan are positive
Practise this
New questions every time.
Common Core-referenced: HSF-TF.B.7