3. Unit Circle

Without using a calculator, determine which of the two values is greater.

tan 1 or tan 2

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = 2π

(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it has a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure. S and L are in feet. The minimum length L of a sag curve is determined by the height h of the car's headlights above the pavement, the downhill grade θ₁ < 0°, the uphill grade θ₂ > 0°, and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle α above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula (θ₂ - θ₁)S² L = ————————— , 200(h + S tan α) where S < L. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) Compute length L, to the nearest foot, if h = 1.9 ft, α = 0.9°, θ₁ = -3°, θ₂ = 4°, and S = 336 ft.

Use a calculator to evaluate each expression. sin 35° cos 55° + cos 35° sin 55°

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin² 𝜋 + cos² 𝜋 10 10

Without using a calculator, determine which of the two values is greater.

cos 2 or sin 2

Determine whether each statement is true or false. See Example 4. csc 20° < csc 30°

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 6, c = 7

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

[-2π , π) ; 3 tan² s = 1

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. tan² 72°25' + 1 = sec² 72°25'

Find each exact function value.

tan π/3

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. ½ sin 40° = sin [½ (40°)]

In Exercises 25–32, 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.

<Image>

sin 47𝜋/4

Find the sine, cosine, and tangent of each angle using the unit circle.

$\(\theta\)=225\(\degree\),\(\left\)(-\(\frac{\sqrt2}{2}\),-\(\frac{\sqrt2}{2}\]\right\))$