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Chapter 2: Graphing Trigonometric Functions

Difficulty Level: At Grade Created by: CK-12

Chapter Outline

Chapter Summary

In this chapter we learned about another way to measure degrees, called radians. Radians are based off of the unit circle, which is a circle with a radius of one. Because all circles are similar, it doesn’t matter which one we use to measure radians, the ratios will always reduce to be the same. Therefore, we can now graph the six trigonometric functions on the xy\begin{align*}x-y\end{align*} plane. All the trigonometric functions are periodic and, wherever the trig value is undefined the graph has a vertical asymptote. Finally, we learned that trig functions can be transformed, much like a quadratic or cubic function. Every trig function has amplitude, phase shift, vertical shift and a period, or frequency.

Vocabulary

Amplitude
A dilation on the y\begin{align*}y-\end{align*}value of a trigonometric function. A\begin{align*}A\end{align*} is multiplied by the function, to make it stretch or shorten.
Angular Velocity
The velocity of rotation, measured in radians.
Arc Length
The length of a portion of the circumference of a circle. The formula is θ2πr\begin{align*}\theta \cdot 2\pi r\end{align*}, where θ\begin{align*}\theta\end{align*} is the corresponding central angle, in radians.
Circular Function
The collective term for a function that can be defined by the unit circle.
Critical Angle(s)
Any angle that is a multiple of 30\begin{align*}30^\circ\end{align*} or 45\begin{align*}45^\circ\end{align*}.
Dilation
A transformation that changes the size of an object or function.
Frequency
The number of times the graph repeats in 2π\begin{align*}2\pi\end{align*} or π\begin{align*}\pi\end{align*} for tangent and cotangent.
Period
The distance it takes a graph to complete one phase.
Phase Shift
The shift, or translation, in the x\begin{align*}x-\end{align*}direction of a trigonometric function. Also called a horizontal translation.
An alternate way to measure degrees, defined by the arc length on a circle that is equal to the radius. 360=2π\begin{align*}360^\circ = 2\pi\end{align*} radians.
Sector
The area of a portion of a circle. The formula is θπr2\begin{align*}\theta \cdot \pi r^2\end{align*}, where θ\begin{align*}\theta\end{align*} is the central angle, measured in radians.
Transformation
Any change made to an object or graph. Transformations can either be dilations or translations.
Translation
Either a vertical or horizontal movement of an object or function.
Vertical Shift
The vertical translation of a function.

Review Questions

1. Convert 160\begin{align*}160^\circ\end{align*} to radians.
2. Convert 11π12\begin{align*}\frac{11\pi}{12}\end{align*} to degrees.
3. Find the exact value of cos3π4\begin{align*}\cos \frac{3\pi}{4}\end{align*}.
4. Find all possible answers in radians, between 0<θ<2π:tanθ=3\begin{align*}0 < \theta < 2\pi: \tan \theta =\sqrt{3}\end{align*}
5. This is an image of the state flag of Colorado

It turns out that the diameter of the gold circle is 13\begin{align*}\frac{1}{3}\end{align*} the total height of the flag (the same width as the white stripe) and the outer diameter of the red circle is 23\begin{align*}\frac{2}{3}\end{align*} of the total height of the flag. The angle formed by the missing portion of the red band is π4\begin{align*}\frac{\pi}{4}\end{align*} radians. In a flag that is 66 inches tall, what is the area of the red portion of the flag to the nearest square inch?

6. Suppose the radius of the dial of an electric meter on a house is 7 cm.

1. How fast is a point on the outside edge of the dial moving if it completes a revolution in 9 seconds?
2. Find the angular velocity of a point on the dial.
7. In the figure below, there is a quadrilateral formed by four line segments: the two radii of the circle (in pink), the orange segment (marked as “cotangent”), and the green segment (marked as “tangent”). The tangent (green) segment has been constructed as tangent to the circle (forming a 90-degree angle with the radius). How do you know that the number of units that is the length of the cotangent segment is equal to xy\begin{align*}\frac{x}{y}\end{align*}? You may assume that the radii shown (pink) are 1 unit.
8. Graph y=sinx\begin{align*}y=\sin x\end{align*} and y=cosx\begin{align*}y=\cos x\end{align*} on the same set of axes over the interval [0,2π]\begin{align*}[0, 2\pi]\end{align*}. Where do they intersect?

For questions 9-12, determine the amplitude, period, frequency, vertical shift, and phase shift. Then, graph each function over the interval [0,2π]\begin{align*}[0, 2\pi]\end{align*}.

1. y=2+4sin5x\begin{align*}y=-2+4 \sin 5x\end{align*}
2. f(x)=14cos(12(xπ3))\begin{align*}f(x)=\frac{1}{4} \cos \left(\frac{1}{2}(x-\frac{\pi}{3}\right))\end{align*}
3. g(x)=4+tan(2(x+π2))\begin{align*}g(x)=4+\tan \left(2(x+\frac{\pi}{2}\right))\end{align*}
4. h(x)=36cos(πx)\begin{align*}h(x)=3-6 \cos (\pi x)\end{align*}

For questions 13 and 14, find the equation of the graph below. Only sine and cosine functions will have an amplitude other than 1.

Texas Instruments Resources

In the CK-12 Texas Instruments Trigonometry FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9700.

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