Express

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°)

Answer 2

To express 4 cos 𝑥 + 3 sin 𝑥 in the form Rsin(𝑥 + 𝜃), we can use the following steps:

Step 1: Rewrite the expression using the identity: 𝑅sin(𝑥 + 𝜃) = 𝑅(sin 𝑥 cos 𝜃 + cos 𝑥 sin 𝜃).

Step 2: Rearrange the terms to match the given expression: 𝑅(sin 𝑥 cos 𝜃 + cos 𝑥 sin 𝜃) = 𝑅cos 𝜃 sin 𝑥 + 𝑅sin 𝜃 cos 𝑥.

Step 3: Compare the coefficients of sin 𝑥 and cos 𝑥 in the given expression with 𝑅sin 𝜃 and 𝑅cos 𝜃 from Step 2. Equate the coefficients to find 𝑅sin 𝜃 and 𝑅cos 𝜃.

From the given expression, we have:
4 cos 𝑥 + 3 sin 𝑥 = 𝑅cos 𝜃 sin 𝑥 + 𝑅sin 𝜃 cos 𝑥.

Comparing the coefficients, we can set up the following equations:
𝑅cos 𝜃 = 4
𝑅sin 𝜃 = 3

Step 4: Solve the equations from Step 3 to find 𝑅, 𝜃.

Dividing the second equation by the first equation, we get:
(𝑅sin 𝜃) / (𝑅cos 𝜃) = 3 / 4
tan 𝜃 = 3 / 4

Taking the arctan of both sides to solve for 𝜃:
𝜃 = arctan(3 / 4)

Substituting the value of 𝜃 into the first equation, we can solve for 𝑅:
𝑅cos (arctan(3 / 4)) = 4

Using the identity: cos(arctan(𝑥)) = 1 / sqrt(1 + 𝑥^2), we can rewrite the equation as:
𝑅 / sqrt(1 + (3 / 4)^2) = 4
𝑅 / sqrt(1 + 9 / 16) = 4
𝑅 / sqrt(25 / 16) = 4
𝑅 / (5 / 4) = 4
𝑅 = 4 * (5 / 4)
𝑅 = 5

Step 5: Substitute the values of 𝑅 and 𝜃 back into the expression 𝑅sin(𝑥 + 𝜃):

𝑅sin(𝑥 + 𝜃) = 5sin(𝑥 + arctan(3 / 4))

So, the expression 4 cos 𝑥 + 3 sin 𝑥 can be written in the form 5sin(𝑥 + arctan(3 / 4)).