# 3.4.1: Exercises 3.4

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## Terms and Concepts

Exercise $$\PageIndex{1}$$

Explain why the domain of tangent is restricted.

Tangent is undefined whenever cosine is 0.

Exercise $$\PageIndex{2}$$

Explain why the domain of cosecant is restricted.

Cosecant is undefined whenever sine is 0.

Exercise $$\PageIndex{3}$$

Explain what is meant by the range of a function.

The range describes the possible output values of the function.

Exercise $$\PageIndex{4}$$

What do the coordinates on the unit circle tell you?

The x coordinate tells you the value of cosine for that angle and the y coordinate tells you the value of sine for that angle.

Exercise $$\PageIndex{5}$$

Sketch the unit circle from memory. Use Figure 3.4.2 to check your work and add in any values you could not remember.

## Problems

Evaluate each statement given in exercises $$\PageIndex{6}$$ - $$\PageIndex{10}$$.

Exercise $$\PageIndex{6}$$

$$\displaystyle \tan{\bigg( \frac{\pi}{4}\bigg)}$$

$$\displaystyle 1$$

Exercise $$\PageIndex{7}$$

$$\displaystyle \cos{\bigg( \frac{-\pi}{4}\bigg)}$$

$$\displaystyle \frac{\sqrt{2}}{2}$$

Exercise $$\PageIndex{8}$$

$$\displaystyle \sin{\bigg( \frac{3\pi}{4}\bigg)}$$

$$\displaystyle \frac{\sqrt{2}}{2}$$

Exercise $$\PageIndex{9}$$

$$\displaystyle \csc{\bigg( \frac{-3\pi}{4}\bigg)}$$

$$\displaystyle -\sqrt{2}$$

Exercise $$\PageIndex{10}$$

$$\displaystyle \sin{\bigg( \frac{3\pi}{2}\bigg)}$$

$$\displaystyle -1$$

Determine the range of each function given in exercises $$\PageIndex{11}$$ - $$\PageIndex{14}$$.

Exercise $$\PageIndex{11}$$

$$f(x) = -2 \sin{(4x)} + 3$$

$$[1,5]$$

Exercise $$\PageIndex{12}$$

$$g(x) = 6\cos{(2x)} -8$$

$$[-14,-2]$$

Exercise $$\PageIndex{13}$$

$$h(x) = -\sin{(x)} -1$$

$$[-2,0]$$

Exercise $$\PageIndex{14}$$

$$f(\theta) =4\sin{(\theta-\pi)}$$

$$[-4,4]$$

In exercises $$\PageIndex{15}$$ - $$\PageIndex{18}$$, use the unit circle to help you answer the given question.

Exercise $$\PageIndex{15}$$

Find the ordered pair for the point on the unit circle associated with $$\theta=\frac{5\pi}{4}$$

$$(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$$

Exercise $$\PageIndex{16}$$

Sketch the a unit circle and the angle represented by $$\theta = \frac{7\pi}{6}$$. Find the ordered pair where this line intersects the unit circle and label this point on your sketch.

Exercise $$\PageIndex{17}$$

Sketch the a unit circle and the angle represented by $$\theta = -\frac{2\pi}{3}$$. Find the ordered pair where this line intersects the unit circle and label this point on your sketch.

Exercise $$\PageIndex{18}$$
Find the equation of the line that intersects the unit circle at $$\theta = \pi$$ and at $$\theta=\frac{\pi}{3}$$. Answer in slope intercept form.
$$y=\frac{\sqrt{3}}{3} x+\frac{\sqrt{3}}{3}$$