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Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function:
In [link] , the constant $\alpha $ causes a horizontal or phase shift. The factor $B$ changes the period. This transformed sine function will have a period $2\pi \text{/}\left|B\right|.$ The factor $A$ results in a vertical stretch by a factor of $\left|A\right|.$ We say $\left|A\right|$ is the “amplitude of $f.$ ” The constant $C$ causes a vertical shift.
Notice in [link] that the graph of $y=\text{cos}\phantom{\rule{0.1em}{0ex}}x$ is the graph of $y=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ shifted to the left $\pi \text{/}2$ units. Therefore, we can write $\text{cos}\phantom{\rule{0.1em}{0ex}}x=\text{sin}(x+\pi \text{/}2).$ Similarly, we can view the graph of $y=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ as the graph of $y=\text{cos}\phantom{\rule{0.1em}{0ex}}x$ shifted right $\pi \text{/}2$ units, and state that $\text{sin}\phantom{\rule{0.1em}{0ex}}x=\text{cos}(x-\pi \text{/}2).$
A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with $15.7$ hours and December 21 is the shortest day of the year with $8.3$ hours. It can be shown that the function
is a model for the number of hours of daylight $h$ as a function of day of the year $t$ ( [link] ).
Sketch a graph of $f\left(x\right)=3\phantom{\rule{0.1em}{0ex}}\text{sin}\left(2\left(x-\frac{\pi}{4}\right)\right)+1.$
This graph is a phase shift of $y=\text{sin}\left(x\right)$ to the right by $\pi \text{/}4$ units, followed by a horizontal compression by a factor of 2, a vertical stretch by a factor of 3, and then a vertical shift by 1 unit. The period of $f$ is $\pi .$
Describe the relationship between the graph of $f\left(x\right)=3\phantom{\rule{0.1em}{0ex}}\text{sin}\left(4x\right)-5$ and the graph of $y=\text{sin}(x).$
To graph $f\left(x\right)=3\phantom{\rule{0.1em}{0ex}}\text{sin}\left(4x\right)-5,$ the graph of $y=\text{sin}(x)$ needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function $f$ will have a period of $\pi \text{/}2$ and an amplitude of 3.
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $\pi .$
$240\text{\xb0}$
$\frac{4\pi}{3}\phantom{\rule{0.2em}{0ex}}\text{rad}$
$15\text{\xb0}$
$\mathrm{-225}\text{\xb0}$
$330\text{\xb0}$
$\frac{11\pi}{6}\phantom{\rule{0.2em}{0ex}}\text{rad}$
For the following exercises, convert each angle in radians to degrees.
$\frac{\pi}{2}\phantom{\rule{0.2em}{0ex}}\text{rad}$
$\frac{7\pi}{6}\phantom{\rule{0.2em}{0ex}}\text{rad}$
$210\text{\xb0}$
$\frac{11\pi}{2}\phantom{\rule{0.2em}{0ex}}\text{rad}$
$\mathrm{-3}\pi \phantom{\rule{0.1em}{0ex}}\text{rad}$
$\mathrm{-540}\text{\xb0}$
$\frac{5\pi}{12}\phantom{\rule{0.2em}{0ex}}\text{rad}$
Evaluate the following functional values.
$\text{tan}\left(\frac{19\pi}{4}\right)$
$\text{sin}\left(-\frac{3\pi}{4}\right)$
$-\frac{\sqrt{2}}{2}$
$\text{sec}\left(\frac{\pi}{6}\right)$
$\text{sin}\left(\frac{\pi}{12}\right)$
$\frac{\sqrt{3}-1}{2\sqrt{2}}$
$\text{cos}\left(\frac{5\pi}{12}\right)$
For the following exercises, consider triangle ABC , a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A . Where necessary, round to one decimal place.
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