Answer:
cos(x + π/6) = (√3/2)cos(x) - (1/2)sin(x).
Explanation:
To rewrite cos(x + π/6) in terms of sine(x) and cos(x) without explicitly using π/6, we can utilize the sum or difference formula for cosine.
The sum formula for cosine states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
In this case, let's consider A = x and B = π/6. Using the sum formula, we have:
cos(x + π/6) = cos(x)cos(π/6) - sin(x)sin(π/6).
Now, we can simplify further. The value of cos(π/6) and sin(π/6) can be determined using the unit circle or trigonometric identities.
cos(π/6) = √3/2 and sin(π/6) = 1/2.
Substituting these values into the equation, we get:
cos(x + π/6) = cos(x)(√3/2) - sin(x)(1/2).
Thus, cos(x + π/6) can be expressed in terms of sine(x) and cos(x) as:
cos(x + π/6) = (√3/2)cos(x) - (1/2)sin(x).