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)