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.
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sin 3𝜋/4

Verified step by step guidance

1. 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.
2. 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 \( (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \).
3. 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 \( \frac{\sqrt{2}}{2} \).
4. 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 \( k \).
5. 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.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of these trigonometric functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, the sine of an angle is the y-coordinate, while the cosine is the x-coordinate of the corresponding point on the circle. Understanding these functions is crucial for solving problems involving angles and their relationships in various contexts, including periodic properties.

Periodic Properties

Periodic properties refer to the repeating nature of trigonometric functions. For example, the sine and cosine functions have a period of 2π, meaning their values repeat every 2π radians. This property allows us to find the values of trigonometric functions for angles greater than 2π or less than 0 by adding or subtracting multiples of 2π, which is essential for evaluating functions at various angles.