If a cos theta plus b sin theta equals 8 and a sin theta - b cos theta equals 5 show that a squared plus b squared equals 89?

Answer 1

There is a hint to how to solve this in what is required to be shown: a and b are both squared.

If

1. a cos θ + b sin θ = 8
2. a sin θ - b cos θ = 5

then square both sides of each to get:

1. a² cos² θ + 2ab cos θ sin θ + b² sin² θ = 64
2. a² sin² θ - 2ab sin θ cos θ + b² cos² θ = 25

Now add the two together:

a² cos² θ + a² sin² θ + b² sin² θ + b² cos² θ = 89

→ a²(cos² θ + sin² θ) + b² (sin² θ + cos² θ) = 89

using cos² θ + sin² θ = 1

→ a² + b² = 89