Express

4 cos π‘₯ + 3 sin π‘₯ in the
form Rsin(π‘₯ + πœƒ)

1 answer

First, let us use the identity 𝑅sin(π‘₯+πœƒ) = 𝑅cos(πœƒ)sin(π‘₯) + 𝑅sin(πœƒ)cos(π‘₯) to rewrite the expression in the desired form.

We can rewrite 4 cos π‘₯ + 3 sin π‘₯ as:

4 cos π‘₯ + 3 sin π‘₯ = 𝑅cos(πœƒ)sin(π‘₯) + 𝑅sin(πœƒ)cos(π‘₯)

where 𝑅 and πœƒ are constants that we need to determine.

To find 𝑅 and πœƒ, we can use the following relationships:

𝑅 = √(4^2 + 3^2) = 5

tan πœƒ = 3/4

πœƒ β‰ˆ 36.87Β°

Therefore, we have:

4 cos π‘₯ + 3 sin π‘₯ = 5 cos(36.87Β°) sin π‘₯ + 5 sin(36.87Β°) cos π‘₯

So, the expression in the form Rsin(π‘₯ + πœƒ) is:

5 sin(π‘₯ + 36.87Β°)
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