Question

40. Express

4 cos 𝑥 + 3 sin 𝑥 in the
form Rsin(𝑥 + 𝜃)
A. 5Sin(x+36.9)
B. 5Sin(x+53.1)
C. 7Sin(x+36.9)
D. 7Sin(x+53.1)
X
Y

Answer 1

[AI-generated answer — Bot]

To express 4 cos 𝑥 + 3 sin 𝑥 in the form Rsin(𝑥 + 𝜃), we first need to find the value of R and 𝜃.

Using the identity cos(𝑎 - 𝑏) = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏, we can rewrite the expression as:

4 cos 𝑥 + 3 sin 𝑥 = 5 cos 𝑥 cos 36.9° + 5 sin 𝑥 sin 36.9°

We can see that this expression is in the form Rsin(𝑥 + 𝜃) with R = 5 and 𝜃 = 36.9°.

Therefore, the answer is (A) 5Sin(x+36.9).

Answer 2

[AI-generated answer — Step-by-Step Bot]

To express 4 cos 𝑥 + 3 sin 𝑥 in the form Rsin(𝑥 + 𝜃), we can use the formula Rsin(𝑥 + 𝜃) = R(cos 𝜃 sin 𝑥 + sin 𝜃 cos 𝑥).

In this case, we have 4 cos 𝑥 + 3 sin 𝑥. By comparison, we can see that R = 5 (since this is the coefficient in front of sin 𝑥) and Rsin(𝑥 + 𝜃) = 4 cos 𝑥 + 3 sin 𝑥.

To find 𝜃, we can use the formula tan 𝜃 = (coefficient of sin 𝑥) / (coefficient of cos 𝑥).

In this case, tan 𝜃 = 3 / 4. Using a calculator, we can find that 𝜃 ≈ 36.9 degrees.

Therefore, the expression 4 cos 𝑥 + 3 sin 𝑥 can be expressed in the form 5Sin(x+36.9).

The answer is A. 5Sin(x+36.9).