Question

Consider the following formulas.

a sin Bθ + b cos Bθ = a2 + b2 sin(Bθ + C), where C = arctan(b/a) and a > 0
a sin Bθ + b cos Bθ = a2 + b2 cos(Bθ − C), where C = arctan(a/b) and b > 0
Use the formulas given above to write the trigonometric expression in the form a sin Bθ + b cos Bθ.
11 cos (θ − π/ 4)

Answer 1

To write 11 cos (θ − π/4) in the form a sin Bθ + b cos Bθ, use the formula a sin Bθ + b cos Bθ = a² + b² cos(Bθ − C), where C = arctan(b/a) and a > 0. Substitute the values a = 11, B = 1, and C = arctan(1/11) into the formula to simplify the expression.

Step-by-step explanation:

To write the trigonometric expression 11 cos (θ − π/4) in the form a sin Bθ + b cos Bθ, we can use the formula a sin Bθ + b cos Bθ = a² + b² sin(Bθ + C), where C = arctan(b/a) and a > 0.

In this case, a = 11, B = 1, and C = arctan(1/11).

Since the given expression is in the form cos(θ − φ), which matches the pattern cos(Bθ − C), we can use the formula a sin Bθ + b cos Bθ = a² + b² cos(Bθ − C) to simplify it.

Substituting the values, we have 11 cos (θ − π/4) = 11² + 1² cos(θ − arctan(1/11)).