M5.3.4 For the adult
Solving trig equations
If sin x = 0.4, what is x, and why are there so many answers?
What this chapter teaches
An equation such as sin x = 0.4 has many solutions, because the wave reaches each height again and again. To solve it in an interval: find the acute reference angle with the calculator, use the quadrants where the sign is right, and add or take away full turns. For sin 2x, solve in a doubled interval first. For a quadratic in sin x, factorise rather than divide, so no solution is lost.
Three questions to ask
There is no one right wording. Listen for the idea.
- Why does sin x = 0.4 have two solutions between 0 and 2π?
- Why must the interval be changed for sin 2x before you solve?
- What can go wrong if you divide both sides of an equation by cos x?
After the chapter, the learner can
- Solve sin x = 0.4 for 0 ≤ x ≤ 2π
- Solve 2cos²x − cos x − 1 = 0
What they copy into their notebook
Reference angle α from |k|. Choose quadrants by CAST. Add full turns. For sin 2x, double the interval first. Factorise, never divide by a trig function that can be 0.