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Find the following trigonometric values.
Express your answers exactly.

{:[cos((7pi)/(4))=◻],[sin((7pi)/(4))=◻]:}

Find the following trigonometric values. $\newline$ Express your answers exactly. $\newline$ $\begin{array}{l} \cos \left(\frac{7 \pi}{4}\right)=\square \\ \sin \left(\frac{7 \pi}{4}\right)=\square \end{array}$

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Write variable expressions: word problems

Full solution

Q. Find the following trigonometric values.

\newline

Express your answers exactly.

\newline

\begin{array}{l} \cos \left(\frac{7 \pi}{4}\right)=\square \\ \sin \left(\frac{7 \pi}{4}\right)=\square \end{array}

Finding Exact Values: We need to find the exact values of the cosine and sine functions for the angle $(7\pi)/4$ . This angle corresponds to a standard position on the unit circle, which is $45^\circ$ (or $\pi/4$ radians) past the $3\pi/2$ position (or $270^\circ$ ), placing it in the fourth quadrant.
Quadrant and Coordinates: In the fourth quadrant, the cosine value is positive and the sine value is negative. This is because the cosine corresponds to the x-coordinate and the sine corresponds to the y-coordinate of a point on the unit circle.
Reference Angle: The reference angle for $(7\pi)/4$ is $\pi/4$ because $(7\pi)/4$ is an angle that is $\pi/4$ radians past the $3\pi/2$ position. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.
Cosine and Sine Values: The cosine and sine of $\frac{\pi}{4}$ are both $\frac{\sqrt{2}}{2}$ . Since $\frac{7\pi}{4}$ is in the fourth quadrant, we must change the sign of the sine value to negative. Therefore, $\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$ and $\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ .

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