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Chapter 1: Problem 8

What is the domain of the secant function?

Short Answer

Expert verified
Answer: The domain of the secant function is all real numbers x except the odd multiples of 饾懃=\dfrac{(2n+1)\pi}{2}, where n is any integer. In interval notation, the domain of the secant function can be represented as: (-鈭�, 饾懃=\dfrac{\pi}{2}) 鈭� ( 饾懃=\dfrac{\pi}{2}, 饾懃=\dfrac{3\pi}{2}) 鈭� ( 饾懃=\dfrac{3\pi}{2}, 饾懃=\dfrac{5\pi}{2}) 鈭� ...

Step by step solution

01

Identify the secant function relationship with cosine function

We know that the secant function is the reciprocal of the cosine function: sec(x) = 1/cos(x).
02

Identify undefined values for cosine function

Cosine function only becomes undefined when its value is equal to zero, so we need to find the values of x when cosine(x) = 0.
03

Find the values of x where cosine(x) = 0

For the cosine function, the value becomes zero at odd multiples of \dfrac{\pi}{2}. That is, when x = \dfrac{(2n+1)\pi}{2}, where n is any integer.
04

Formulate the domain of the secant function

Since the secant function is undefined at the values of x mentioned in Step 3, the domain of the secant function is given by all real numbers x except the odd multiples of \dfrac{\pi}{2}. In interval notation, the domain of the secant function can be represented as: (-\infty, \dfrac{\pi}{2}) \cup (\dfrac{\pi}{2}, \dfrac{3\pi}{2}) \cup (\dfrac{3\pi}{2}, \dfrac{5\pi}{2}) \cup ...

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Most popular questions from this chapter

An auditorium with a flat floor has a large flatpanel television on one wall. The lower edge of the television is \(3 \mathrm{ft}\) above the floor, and the upper edge is \(10 \mathrm{ft}\) above the floor (see figure). Express \(\theta\) in terms of \(x\)

Determine whether the graphs of the following equations and functions have symmetry about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$f(x)=x^{4}+5 x^{2}-12$$

a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. b. Give the domain of the function. c. Discuss the interesting features of the function such as peaks, valleys, and intercepts (as in Example 5 ). $$g(x)=\left|\frac{x^{2}-4}{x+3}\right|$$

(Torricelli's law) A cylindrical tank with a cross-sectional area of \(100 \mathrm{cm}^{2}\) is filled to a depth of \(100 \mathrm{cm}\) with water. At \(t=0,\) a drain in the bottom of the tank with an area of \(10 \mathrm{cm}^{2}\) is opened, allowing water to flow out of the tank. The depth of water in the tank at time \(t \geq 0\) is \(d(t)=(10-2.2 t)^{2}\) a. Check that \(d(0)=100,\) as specified. b. At what time is the tank empty? c. What is an appropriate domain for \(d ?\)

Use shifts and scalings to graph then given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. $$h(x)=-4 x^{2}-4 x+12$$
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