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How can you express sin(x - 5π/6) in terms of sin(x) and cos(x)?

a) -sin(x)cos(5π/6) - cos(x)sin(5π/6)
b) sin(x)cos(5π/6) - cos(x)sin(5π/6)
c) sin(x)cos(5π/6) + cos(x)sin(5π/6)
d) -sin(x)cos(5π/6) + cos(x)sin(5π/6)

User Bob Snyder
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1 Answer

4 votes

Final answer:

The expression sin(x - 5π/6) can be rewritten using the sine addition and subtraction formula as sin(x) cos(5π/6) - cos(x) sin(5π/6), which simplifies to -√3/2 sin(x) - 1/2 cos(x). option b is correct answer.

Step-by-step explanation:

To express sin(x - 5π/6) in terms of sin(x) and cos(x), we can use the sine addition and subtraction formula: sin (a ± β) = sin a cos β ± cos a sin β. For the given expression, we let a = x and β = -5π/6 and apply the formula:

sin(x - 5π/6) = sin(x) cos(-5π/6) + cos(x) sin(-5π/6)

Knowing that cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), we can simplify further:

sin(x - 5π/6) = sin(x) cos(5π/6) - cos(x) sin(5π/6)

The sine of 5π/6 is 1/2 and the cosine of 5π/6 is -√3/2. Plugging these values in, we get:

sin(x - 5π/6) = sin(x) (-√3/2) - cos(x) (1/2)

Now, we multiply through to simplify:

sin(x - 5π/6) = -√3/2 sin(x) - 1/2 cos(x)

Therefore, the correct expression in terms of sin(x) and cos(x) is:

b) sin(x) cos(5π/6) - cos(x) sin(5π/6)

User CephBirk
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