M5.3.8 For the adult
Combining sin and cos
What do I get when I add a sine wave and a cosine wave?
What this chapter teaches
A sum a sin x + b cos x is a single sine wave R sin(x + α). To find it, match the numbers in front of sin x and cos x: R cos α = a and R sin α = b. Then R = √(a² + b²) and tan α = b/a, with the signs of a and b deciding the quadrant of α. The greatest value is R, and an equation a sin x + b cos x = c has solutions only when c is between −R and R.
Three questions to ask
There is no one right wording. Listen for the idea.
- Why is the greatest value of 3 sin x + 4 cos x equal to 5, and not 7?
- Why does tan α = b/a not tell you the quadrant of α?
- What happens to the equation 3 sin x + 4 cos x = 6, and why?
After the chapter, the learner can
- Write 3 sin x + 4 cos x as 5 sin(x + α)
- Find the largest value of 3 sin x + 4 cos x
What they copy into their notebook
a sin x + b cos x = R sin(x + α). R = √(a² + b²). R cos α = a, R sin α = b. Greatest value R. Solve: sin(x + α) = c/R in the interval shifted by α. Needs |c| ≤ R.