# 2.10: Cosine and Secant Graphs

**At Grade**Created by: CK-12

**Practice**Cosine and Secant Graphs

Imagine for a moment that you have a clock that has only one hand - that rotates counterclockwise!. However, the hand is very slim all the way until the tip, where there is a ball on the end. In fact, the hand is so slim you won't notice it. You only notice the ball on the end of the rotating hand. This hand is rotating faster than normal.

Consider what it would be like if you put a light above the clock and let the shadow of the hands fall on the wall under the clock. What pattern would that shadow trace out? If you think about it, you might realize that the shadow would make an left and right motion, over and over as the hand of the clock rotated. Now imagine that instead of a wall, there was a large piece of paper for the shadow to fall on. And wherever the shadow fell, there would be a mark on the paper. Finally, imagine moving the paper as the clock rotates. Can you imagine sort of pattern this would trace out?

By the end of this Concept, you'll understand just how this relates to trigonometric functions in general, and the cosine graph in particular.

### Watch This

James Sousa Animation: Graphing the Cosine Function Using the Unit Circle

### Guidance

If you have read other Trigonometry Concepts in this course, you may have learned that sine and cosine are very closely related. The cosine of an angle is the same as the sine of its complementary angle. So, it should not be a surprise that sine and cosine waves are very similar in that they are both periodic with a period of \begin{align*}2\pi\end{align*}

The cosine of an angle is the ratio of \begin{align*}\frac{x}{r}\end{align*}

Here is a sequence of rotations. Compare the \begin{align*}x-\end{align*}

Plotting the quadrant angles and filling in the in-between values shows the graph of \begin{align*}y = \cos x\end{align*}

The graph of \begin{align*}y = \cos x\end{align*}**range** of a cosine curve is \begin{align*}\left \{-1 \le y \le 1 \right \}\end{align*}**domain** of \begin{align*}\cos x\end{align*}*shifted* by \begin{align*}\frac{\pi}{2}\end{align*}

Secant is the reciprocal of cosine, or \begin{align*}\frac{1}{x}\end{align*}

The period of the function is \begin{align*}2\pi\end{align*}*except* where cosine is defined (other than the tops and bottoms of the cosine curve).

Notice again the reciprocal relationships at 0 and the asymptotes. Also look at the intersection points of the graphs at 1 and -1. Again, this graph looks parabolic, but it is not.

#### Example A

Sketch a graph of \begin{align*}h(x)=5+\frac{1}{2} \sec 4x\end{align*}

**Solution:** If you compare this example to \begin{align*}f(x)=\sec x\end{align*}

#### Example B

Find the equation for the graph below.

**Solution:** First of all, this could be either a secant or cosecant function. Let’s say this is a secant function. Secant usually intersects the \begin{align*}y-\end{align*}

#### Example C

Graph the function \begin{align*}h(x)=2 - 3 \cos 4x\end{align*}

### Vocabulary

**Circular Function:** A ** circular function** is a function that is measured by examining the angle of rotation around the coordinate plane.

### Guided Practice

1. Graph \begin{align*}y=-2+\frac{1}{2} \sec(4(x-1))\end{align*}.

2. Determine the function creating this graph:

3. Graph \begin{align*}h(x)=\frac{1}{3} \cos 2x\end{align*}

**Solutions:**

1.

2. This could be either a secant or cosecant function. We will use a cosecant model. First, the vertical shift is -1. The period is the difference between the two given \begin{align*}x-\end{align*}values, \begin{align*}\frac{7\pi}{4}-\frac{3\pi}{4}=\pi\end{align*}, so the frequency is \begin{align*}\frac{2\pi}{\pi}=2\end{align*}. The horizontal shift incorporates the frequency, so in \begin{align*}y=\csc x\end{align*} the corresponding \begin{align*}x-\end{align*}value to \begin{align*}\left(\frac{3\pi}{4}, 0\right)\end{align*} is \begin{align*}\left(\frac{\pi}{2}, 1\right)\end{align*}. The difference between the \begin{align*}x-\end{align*}values is \begin{align*}\frac{3\pi}{4}-\frac{\pi}{2}=\frac{3\pi}{4}-\frac{2\pi}{4}=\frac{\pi}{4}\end{align*} and then multiply it by the frequency, \begin{align*}2 \cdot \frac{\pi}{4}=\frac{\pi}{2}\end{align*}. The equation is \begin{align*}y=-1+ \csc \left(2(x-\frac{\pi}{2}\right))\end{align*}.

3.

### Concept Problem Solution

As you have learned in this Concept, a light shining down on the rotating hand would create a shadow in the pattern of a cosine function, starting at a maximum value as the hand is lying along the "x" axis, going through zero to a maximum negative value when the hand is lying along the negative "y" axis. It would then begin to increase until it returned to a maximum value when the rotating hand was again lying along the positive "x" axis.

### Practice

Graph each of the following functions.

- \begin{align*}f(x)=\cos(x)\end{align*}.
- \begin{align*}h(x)=\cos(2x)\end{align*}.
- \begin{align*}k(x)=\cos(2x+\pi)\end{align*}.
- \begin{align*}m(x)=-2\cos(2x+\pi)\end{align*}.
- \begin{align*}g(x)=-2\cos(2x+\pi)+1\end{align*}.
- \begin{align*}f(x)=\sec(x)\end{align*}.
- \begin{align*}h(x)=\sec(3x)\end{align*}.
- \begin{align*}k(x)=\sec(3x+\pi)\end{align*}.
- \begin{align*}m(x)=2\sec(3x+\pi)\end{align*}.
- \begin{align*}g(x)=3+2\sec(3x+\pi)\end{align*}.
- \begin{align*}h(x)=\cos(\frac{x}{2})\end{align*}.
- \begin{align*}k(x)=\cos(\frac{x}{2}+\frac{\pi}{2})\end{align*}.
- \begin{align*}m(x)=2\cos(\frac{x}{2}+\frac{\pi}{2})\end{align*}.
- \begin{align*}g(x)=2\cos(\frac{x}{2}+\frac{\pi}{2})-3\end{align*}.
- \begin{align*}h(x)=\sec(\frac{x}{4})\end{align*}.
- \begin{align*}k(x)=\sec(\frac{x}{4}+\frac{3\pi}{2})\end{align*}.
- \begin{align*}m(x)=-3\sec(\frac{x}{4}+\frac{3\pi}{2})\end{align*}.
- \begin{align*}g(x)=2-3\sec(\frac{x}{4}+\frac{3\pi}{2})\end{align*}.

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### Image Attributions

Here you'll learn to draw the graphs of the cosine and secant functions.