Question

The unit circle has been divided into eight equal arcs, corresponding to t-values of

0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.

a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
<IMAGE>

sin 3𝜋/4

Accepted Answer

Step 1: Understand the unit circle and its coordinates. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. Each point on the unit circle corresponds to an angle $$ t $$ measured in radians from the positive x-axis. Step 2: Identify the coordinates for $$ t = \frac{3\pi}{4} . On $$the unit circle, the angle $$ \frac{3\pi}{4}  is in $$the second quadrant. The coordinates for this angle are $$ (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) . $$Step 3: Determine the sine value. The sine of an angle $$ t  on $$the unit circle is the y-coordinate of the corresponding point. For $$ t = \frac{3\pi}{4} $$, the sine value is $$ \frac{\sqrt{2}}{2} . $$Step 4: Use periodic properties to find the sine value at a different angle. The sine function is periodic with a period of $$ 2\pi . $$This means $$ \sin(t + 2\pi k) = \sin(t) $$ for any integer $$ k . $$Step 5: Apply the periodic property to find the sine value at the indicated real number. If the problem specifies a different angle, use the periodic property to relate it back to $$ \frac{3\pi}{4} $$ and find the sine value.

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Key Concepts

Unit Circle

Trigonometric Functions

Periodic Properties