3. Unit Circle

For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.

s = 65

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

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.

tan 0

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.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3

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