Question

Express
4 cos 𝑥 + 3 sin 𝑥 in the
form Rsin(𝑥 + 𝜃)

Answer 1

First, let us use the identity 𝑅sin(𝑥+𝜃) = 𝑅cos(𝜃)sin(𝑥) + 𝑅sin(𝜃)cos(𝑥) to rewrite the expression in the desired form.

We can rewrite 4 cos 𝑥 + 3 sin 𝑥 as:

4 cos 𝑥 + 3 sin 𝑥 = 𝑅cos(𝜃)sin(𝑥) + 𝑅sin(𝜃)cos(𝑥)

where 𝑅 and 𝜃 are constants that we need to determine.

To find 𝑅 and 𝜃, we can use the following relationships:

𝑅 = √(4^2 + 3^2) = 5

tan 𝜃 = 3/4

𝜃 ≈ 36.87°

Therefore, we have:

4 cos 𝑥 + 3 sin 𝑥 = 5 cos(36.87°) sin 𝑥 + 5 sin(36.87°) cos 𝑥

So, the expression in the form Rsin(𝑥 + 𝜃) is:

5 sin(𝑥 + 36.87°)