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QuestionFind the exact value of sin⁡(−855∘)\sin(-855^\circ). Simplify your answer, using integers or fractions.

Studdy Solution

STEP 1

Assumptions1. We are working with degrees, not radians. . We are looking for the exact value of the sine function for the angle -855 degrees.
3. We know that the sine function is periodic with a period of360 degrees, meaning that sin⁡(θ+360n)=sin⁡(θ)\sin(\theta +360n) = \sin(\theta) for any integer n.

STEP 2

First, we need to find an equivalent angle for -855 degrees that lies within the range of0 to360 degrees. We can do this by adding multiples of360 degrees until we get an angle in this range.
θ′=−855+360n\theta' = -855 +360n

STEP 3

Now, we need to find the smallest positive integer n such that θ′\theta' lies within the range of0 to360 degrees.

STEP 4

We can start by trying n =1.
θ′=−855+360×1=−495\theta' = -855 +360 \times1 = -495This is still not within the range of0 to360 degrees, so we try n =2.
θ′=−855+360×2=−135\theta' = -855 +360 \times2 = -135This is still not within the range of0 to360 degrees, so we try n =3.
θ′=−855+360×3=225\theta' = -855 +360 \times3 =225Now we have an angle that lies within the range of0 to360 degrees.

STEP 5

Now that we have an equivalent angle within the range of0 to360 degrees, we can find the sine of this angle. We know that sin⁡(θ′)=sin⁡(−855)\sin(\theta') = \sin(-855).
sin⁡(−855)=sin⁡(225)\sin(-855) = \sin(225)

STEP 6

The sine of225 degrees can be found using the unit circle or a reference triangle. We know that sin⁡(225)=−2/2\sin(225) = -\sqrt{2}/2.
So, the exact value of the expression issin⁡(−855)=−2/2\sin(-855) = -\sqrt{2}/2

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