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QuestionFind the exact value of f(a)+f(a−4π)+f(a−2π)f(a) + f(a - 4\pi) + f(a - 2\pi) if f(x)=sin⁡xf(x) = \sin x and f(a)=−19f(a) = -\frac{1}{9}.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is defined as f(x)=sin⁡xf(x) = \sin x . The value of f(a)f(a) is given as −19-\frac{1}{9}
3. We are asked to find the exact value of f(a)+f(a−4π)+f(a−π)f(a) + f(a -4\pi) + f(a -\pi)

STEP 2

We know that the sine function has a period of 2π2\pi, which means that sin⁡(x+2π)=sin⁡(x)\sin(x +2\pi) = \sin(x) for any real number xx.

STEP 3

Using the periodic property of the sine function, we can rewrite f(a−2π)f(a -2\pi) and f(a−π)f(a -\pi) as f(a)f(a).
f(a−2π)=sin⁡(a−2π)=sin⁡a=f(a)f(a -2\pi) = \sin(a -2\pi) = \sin a = f(a)f(a−π)=sin⁡(a−π)=sin⁡a=f(a)f(a -\pi) = \sin(a -\pi) = \sin a = f(a)

STEP 4

Substitute f(a)f(a) for f(a−2π)f(a -2\pi) and f(a−4π)f(a -4\pi) in the expression f(a)+f(a−4π)+f(a−2π)f(a) + f(a -4\pi) + f(a -2\pi).
f(a)+f(a−4π)+f(a−2π)=f(a)+f(a)+f(a)=3f(a)f(a) + f(a -4\pi) + f(a -2\pi) = f(a) + f(a) + f(a) =3f(a)

STEP 5

Substitute the given value of f(a)f(a) into the expression 3f(a)3f(a).
3f(a)=3⋅−193f(a) =3 \cdot -\frac{1}{9}

STEP 6

Calculate the final value.
3f(a)=3⋅−19=−133f(a) =3 \cdot -\frac{1}{9} = -\frac{1}{3}So, the exact value of f(a)+f(a−4π)+f(a−2π)f(a) + f(a -4\pi) + f(a -2\pi) is −13-\frac{1}{3}.

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