3. Unit Circle

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.

<IMAGE>

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.

cot 15𝜋/2

Evaluate each expression. See Example 4. cot² 135° - sin 30° + 4 tan 45°

(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye. Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.

If the period of a trigonometric function is $T$, how many complete cycles of the function occur in a horizontal length of $L$?

On the unit circle, if the terminal point is at $(x,y)$, how many radius lengths is it to the right of the circle's vertical diameter?

Given the sequence defined by $a_n=\cos \left(n\pi \right)$, which of the following lists the first five terms of the sequence?

On the unit circle, for $0<\theta <\pi$, when is $tan\left(\theta \right)$ undefined?

Given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ on the unit circle, what is the angle $\theta$ between them if $\overrightarrow{a}$ = $\left(1,0\right)$ and $\overrightarrow{b}$ = $\left(0,1\right)$? Express your answer using one significant figure.

Which of the following statements is true about the measure of an inscribed angle that intercepts an arc on the $unitcircle$?

(Modeling) Speed of Light When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light changes, and the light ray is bent, or refracted, at the boundary between the two media. (This is why objects under water appear to be in a different position from where they really are.) It can be shown in physics that these changes are related by Snell's law c₁ = sin θ₁ , c₂ sin θ₂ where c₁ is the speed of light in the first medium, c₂ is the speed of light in the second medium, and θ₁ and θ₂ are the angles shown in the figure. In Exercises 81 and 82, assume that c₁ = 3 x 10⁸ m per sec. Find the speed of light in the second medium for each of the following. a. θ₁ = 46°, θ₂ = 31° b. θ₁ = 39°, θ₂ = 28°

Given the parametric equation $r\left(t\right)=\left(\cos \right(t),\sin (t),t)$, where does the helix lie?

Figure 10.1 should remind you of trigonometric functions you've seen before. Which of the following functions is most directly represented by the coordinates of a point on the unit circle at an angle $\theta$ from the positive x-axis?

Given the polar equation $r^2=\sin \left(2\theta \right)$, which of the following best describes the shape of its graph on the unit circle?