3.4.1: Exercises 3.4

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May 10, 2021
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- 3.4: Introduction to Trigonometric Functions
- 3.5: Trigonometric Functions and Triangles

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Terms and Concepts

Exercise $\PageIndex{1}$

Explain why the domain of tangent is restricted.

Answer: Tangent is undefined whenever cosine is 0.

Exercise $\PageIndex{2}$

Explain why the domain of cosecant is restricted.

Answer: Cosecant is undefined whenever sine is 0.

Exercise $\PageIndex{3}$

Explain what is meant by the range of a function.

Answer: The range describes the possible output values of the function.

Exercise $\PageIndex{4}$

What do the coordinates on the unit circle tell you?

Answer: 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.

Answer

Problems

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

Exercise $\PageIndex{6}$

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

Answer: $\displaystyle 1$

Exercise $\PageIndex{7}$

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

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

Exercise $\PageIndex{8}$

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

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

Exercise $\PageIndex{9}$

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

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

Exercise $\PageIndex{10}$

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

Answer: $\displaystyle -1$

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

Exercise $\PageIndex{11}$

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

Answer: $[1,5]$

Exercise $\PageIndex{12}$

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

Answer: $[-14,-2]$

Exercise $\PageIndex{13}$

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

Answer: $[-2,0]$

Exercise $\PageIndex{14}$

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

Answer: $[-4,4]$

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 $\theta=\frac{5\pi}{4}$

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

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.

Answer

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.

Answer

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.

Answer: $y=\frac{\sqrt{3}}{3} x+\frac{\sqrt{3}}{3}$

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