3. Unit Circle

Find the exact values of s in the given interval that satisfy the given condition.

[0, 2π) ; sin s = -√3 / 2

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

Concept Check Work each problem. Find the equation of the line that passes through the origin and makes a 30° angle with the x-axis.

Which of the following situations can be modeled with a periodic function?

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.

Which expression is equivalent to $\sin \frac{7\pi }{6}$?

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. csc(β + 40°) = sec(β - 20°)

On the unit circle, what is the radian measure of the arc that subtends a central angle of $90°$?

Find a calculator approximation to four decimal places for each circular function value. cos (-0.2443)

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3 cot θ = - —— 3

Find the exact value of s in the given interval that has the given circular function value.

[π/2, π] ; sin s = 1/2

Find each exact function value. See Example 2. sin 7π/6

Use a calculator to evaluate each expression. 2 sin 25°13' cos 25°13' - sin 50°26'

Find the exact value of s in the given interval that has the given circular function value.

[π, 3π/2] ; tan s = √3

On the unit circle, what relationship do the ratios $\frac{\sin x}{1}$ and $\frac{\cos y}{1}$ share for any real values of $x$ and $y$?