Final answer:
To rewrite the expression 1sin(x) + 6cos(x) as Asin(x + φ), we find A using the Pythagorean theorem and φ using the arctangent of the ratio of the coefficients. The final expression is 6.08sin(x + 1.4056), with A ≈ 6.08 and φ ≈ 1.4056 radians.
Step-by-step explanation:
Expressing a Linear Combination of Sine and Cosine as a Single Sine Function
To rewrite the expression 1sin(x) + 6cos(x) in the form Asin(x + φ), we use the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B). First, we want to express the given linear combination of sine and cosine as a single sine function with a phase shift φ. To do this, we find A and φ such that:
Acos(φ) = 1 (coefficient of sin(x))
Asin(φ) = 6 (coefficient of cos(x))
We calculate the amplitude A using the Pythagorean theorem:
A = √(1^2 + 6^2) = √(1 + 36) = √37 ≈ 6.08
Next, we find the phase shift φ by taking the arctangent of the ratio of the coefficients, which gives us:
φ = atan2(6, 1) ≈ 1.4056 (radians)
However, we need to ensure that φ is in the interval -π < φ < π. The given value of φ is in this interval, so we can just write the final expression as:
6.08sin(x + 1.4056)