1.3 Trigonometric functions  (Page 4/9)

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 $y=\text{cos}x$ is the graph of $y=\text{sin}x$ shifted to the left $\pi \text{/}2$ units. Therefore, we can write $\text{cos}x=\text{sin}(x+\pi \text{/}2).$ Similarly, we can view the graph of $y=\text{sin}x$ as the graph of $y=\text{cos}x$ shifted right $\pi \text{/}2$ units, and state that $\text{sin}x=\text{cos}(x-\pi \text{/}2).$

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] ).

Sketching the graph of a transformed sine curve

Sketch a graph of $f\left(x\right)=3\text{sin}\left(2\left(x-\frac{\pi }{4}\right)\right)+1.$

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 $\pi .$

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Key concepts

• Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of $180\text{°}$ has a radian measure of $\pi$ rad.
• For acute angles $\theta ,$ the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is $\theta .$
• For a general angle $\theta ,$ let $(x,y)$ be a point on a circle of radius $r$ corresponding to this angle $\theta .$ The trigonometric functions can be written as ratios involving $x,y,$ and $r.$
• The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period $2\pi .$ The tangent and cotangent functions have period $\pi .$

Key equations

• Generalized sine function
  $f\left(x\right)=A\text{sin}\left(B\left(x-\alpha \right)\right)+C$

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $\pi .$

For the following exercises, convert each angle in radians to degrees.

Evaluate the following functional values.

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.