In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 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.

cos 9𝜋/2

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Identify the angle 9\(\pi\)/2 on the unit circle. Since the unit circle is periodic with period 2\(\pi\), reduce 9\(\pi\)/2 by subtracting multiples of 2\(\pi\) to find a coterminal angle within the interval [0, 2\(\pi\)).

Calculate the coterminal angle: 9\(\pi\)/2 - 4\(\pi\) = 9\(\pi\)/2 - 8\(\pi\)/2 = \(\pi\)/2. So, 9\(\pi\)/2 is coterminal with \(\pi\)/2.

From the unit circle, find the coordinates corresponding to the angle \(\pi\)/2. The coordinates are (0, 1).

Recall that the cosine of an angle on the unit circle is the x-coordinate of the corresponding point. Therefore, cos(\(\pi\)/2) = 0.

Use the periodic property of cosine, which states that cos(\(\theta\)) = cos(\(\theta\) + 2k\(\pi\)) for any integer k, to conclude that cos(9\(\pi\)/2) = cos(\(\pi\)/2) = 0.

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Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t, measured in radians, and has coordinates (cos t, sin t). These coordinates represent the cosine and sine values of the angle, which are fundamental in trigonometry.

Periodic Properties of Trigonometric Functions

Trigonometric functions like cosine and sine are periodic, meaning their values repeat at regular intervals. For cosine and sine, this period is 2π. This property allows us to find the value of the function at any angle by reducing the angle modulo 2π, simplifying calculations for angles beyond one full rotation.

Special Angles and Their Coordinates

Certain angles on the unit circle, such as π/4, π/2, 3π/4, etc., have well-known coordinates involving square roots, like (√2/2, √2/2). These special angles help in quickly determining the values of trigonometric functions without a calculator, and are essential for solving problems involving exact trigonometric values.