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.

tan 𝜋

Verified step by step guidance

1. Identify the angle given in the problem, which is \(\frac{\pi}{4}\), and locate its corresponding coordinates on the unit circle. From the image, the coordinates for \(\frac{\pi}{4}\) are \(\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)\).
2. Recall that the tangent function is defined as the ratio of the y-coordinate to the x-coordinate on the unit circle, so \(\tan t = \frac{y}{x}\).
3. Using the coordinates for \(\frac{\pi}{4}\), substitute into the tangent formula: \(\tan \frac{\pi}{4} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\).
4. Simplify the fraction to find the value of \(\tan \frac{\pi}{4}\).
5. For part (b), use the periodic property of the tangent function, which has a period of \(\pi\), meaning \(\tan(t + \pi) = \tan t\). Use this property to find the value of the tangent function at the indicated real number by relating it back to the value found in part (a).

Key Concepts

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 (x, y) = (cos t, sin t). These coordinates are essential for evaluating trigonometric functions at specific angles.

Tangent Function on the Unit Circle

The tangent of an angle t is defined as tan t = sin t / cos t, which corresponds to the ratio of the y-coordinate to the x-coordinate of the point on the unit circle. Understanding this ratio helps in finding the value of tangent at given angles using the coordinates from the unit circle.

Periodicity of Trigonometric Functions

Trigonometric functions like tangent are periodic, meaning their values repeat at regular intervals. For tangent, the period is π, so tan(t + π) = tan t. This property allows us to find the value of the tangent function at any angle by relating it to an equivalent angle within one period.