Find the following trigonometric values. Express your answers exactly. {:[cos((7pi)/(4))=◻],[sin((7pi)/(4))=◻]:}

Finding Exact Values: We need to find the exact values of the cosine and sine functions for the angle (7pi)/4. This angle corresponds to a standard position on the unit circle, which is 45 degrees (or pi/4 radians) past the 3pi/2 position (or 270 degrees), 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 (7pi)/4 is pi/4 because (7pi)/4 is an angle that is pi/4 radians past the 3pi/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 pi/4 are both sqrt(2)/2. Since (7pi)/4 is in the fourth quadrant, we must change the sign of the sine value to negative. Therefore, cos((7pi)/4) = sqrt(2)/2 and sin((7pi)/4) = -sqrt(2)/2.

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Write variable expressions: word problems

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Q. Find the following trigonometric values.

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Express your answers exactly.

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\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 $\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ .