In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined. sin 𝜋/3

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 a fundamental tool in trigonometry, as it allows for the definition of sine, cosine, and tangent functions based on the coordinates of points on the circle. The angle in radians corresponds to the rotation from the positive x-axis, and the coordinates (x, y) of any point on the circle represent the cosine and sine of that angle, respectively.

Sine Function

The sine function, denoted as sin(θ), is a trigonometric function that relates an angle θ to the ratio of the length of the opposite side to the hypotenuse in a right triangle. On the unit circle, sin(θ) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. For example, sin(π/3) can be found by identifying the coordinates of the point on the unit circle at that angle.

Radians and Degrees

Radians and degrees are two units for measuring angles. One complete revolution around a circle is 360 degrees, which is equivalent to 2π radians. In trigonometry, radians are often preferred because they simplify the relationships between angles and the lengths of arcs on the unit circle. Understanding how to convert between these two units is essential for solving trigonometric problems, such as finding sin(π/3) accurately.