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Chapter 4: Problem 40

Evaluate the trigonometric function of the quadrant angle, if possible. $$\sec \pi$$

Short Answer

Expert verified
The evaluation of the given trigonometric function \(sec(\pi)\) is -1.

Step by step solution

01

Understanding the Angle

Based on the unit circle, an angle of \(\pi\) radians is equivalent to 180 degrees, which lies in the third quadrant.
02

Evaluating Using Cosine

Evaluate \(\cos(\pi)\), since the secant function is the reciprocal of the cosine function. From knowledge of the unit circle, \(\cos(\pi) = -1\).
03

Evaluation of Given Trigonometric Function

Substitute \(\cos(\pi)\) into the relation \(sec(\theta) = 1/cos(\theta)\) to get \(sec(\pi) = 1/\cos(\pi)\). The expression simplifies to \(sec(\pi) = 1/(-1) = -1\).

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Most popular questions from this chapter

A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5 feet from its low point to its high point (see figure), and it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy where the high point corresponds to the time \(t=0\) (figure cannot copy)

Data Analysis The table shows the average sales \(S\) (in millions of dollars) of an outerwear manufacturer for each month \(t,\) where \(t=1\) represents January. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Sales, } S & 13.46 & 11.15 & 8.00 & 4.85 & 2.54 & 1.70 \\\ \hline \end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Sales, } S & 2.54 & 4.85 & 8.00 & 11.15 & 13.46 & 14.30 \\ \hline \end{array}$$ (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model's amplitude in the context of the problem.

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$f(x)=\frac{1-\cos x}{x}$$

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin \frac{x}{2}$$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$y=\frac{4}{x}+\sin 2 x, \quad x>0$$
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