Check the solution to "Find the exact value of the trigonometric function..."

glamrockqueen7

Answered question

2021-10-29

Find the exact value of the trigonometric function at the given real number.

\sin \frac{7 π}{4}

liannemdh

Skilled2021-10-30Added 106 answers

We would like to find the exact value of

\sin \frac{7 π}{4}

, First, we can find the reference angle of

\frac{7 π}{4}

where

2 π - \frac{7 π}{4} = \frac{π}{4}

, so the reference angle is

\frac{π}{4}

.
Now the next step is to define the sign of

\sin \frac{7 π}{4}

. We know that

\frac{7 π}{4}

is in quadrant 3 which the sine function is negative in this quadrant, so the value of

\sin \frac{7 π}{4}

is negative and we can simplify it as follows:

∴ \sin \frac{7 π}{4} = \sin (2 π - \frac{π}{4}) = - \sin \frac{π}{4}

Note that the first step was to find the reference angle and then was to define the sign of

\sin \frac{7 π}{4}

.
Now we can find the value of

\sin \frac{7 π}{4}

by knowing the value of

\sin \frac{π}{4}

which equals

\frac{\sqrt{2}}{2}

∴ \sin \frac{7 π}{4} = \sin (2 π - \frac{π}{4}) = - \sin \frac{π}{4} = - \frac{\sqrt{2}}{2}

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