3.4.1: Exercises 3.4
- Page ID
- 63466
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Terms and Concepts
Exercise \(\PageIndex{1}\)
Explain why the domain of tangent is restricted.
- Answer
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Tangent is undefined whenever cosine is 0.
Exercise \(\PageIndex{2}\)
Explain why the domain of cosecant is restricted.
- Answer
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Cosecant is undefined whenever sine is 0.
Exercise \(\PageIndex{3}\)
Explain what is meant by the range of a function.
- Answer
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The range describes the possible output values of the function.
Exercise \(\PageIndex{4}\)
What do the coordinates on the unit circle tell you?
- Answer
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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
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Problems
Evaluate each statement given in exercises \(\PageIndex{6}\) - \(\PageIndex{10}\).
Exercise \(\PageIndex{6}\)
\(\displaystyle \tan{\bigg( \frac{\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle 1\)
Exercise \(\PageIndex{7}\)
\(\displaystyle \cos{\bigg( \frac{-\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle \frac{\sqrt{2}}{2}\)
Exercise \(\PageIndex{8}\)
\(\displaystyle \sin{\bigg( \frac{3\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle \frac{\sqrt{2}}{2}\)
Exercise \(\PageIndex{9}\)
\(\displaystyle \csc{\bigg( \frac{-3\pi}{4}\bigg)}\)
- Answer
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\(\displaystyle -\sqrt{2}\)
Exercise \(\PageIndex{10}\)
\(\displaystyle \sin{\bigg( \frac{3\pi}{2}\bigg)}\)
- Answer
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\(\displaystyle -1\)
Determine the range of each function given in exercises \(\PageIndex{11}\) - \(\PageIndex{14}\).
Exercise \(\PageIndex{11}\)
\(f(x) = -2 \sin{(4x)} + 3\)
- Answer
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\([1,5]\)
Exercise \(\PageIndex{12}\)
\(g(x) = 6\cos{(2x)} -8\)
- Answer
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\([-14,-2]\)
Exercise \(\PageIndex{13}\)
\(h(x) = -\sin{(x)} -1\)
- Answer
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\([-2,0]\)
Exercise \(\PageIndex{14}\)
\(f(\theta) =4\sin{(\theta-\pi)}\)
- Answer
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\([-4,4]\)
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
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\((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\)
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
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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
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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
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\(y=\frac{\sqrt{3}}{3} x+\frac{\sqrt{3}}{3}\)