Chapter 11

In \(3-14,\) sketch one cycle of the graph. $$ y=4 \sin (x-\pi) $$

In later courses, you will learn that the cosine function can be written as the sum of an infinite sequence. In particular, for \(x\) in radians, the cosine function can be approximated by the finite series: $$ \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} $$ a. Graph \(Y_{1}=\cos x\) and \(Y_{2}=1-\frac{x^{2}}{2 !}+\frac{x^{2}}{4 !}\) on the graphing calculator. For what values of \(x\) does \(Y_{2}\) seem to be a good approximation for \(Y_{1} ?\) b. The next term of the cosine approximation is \(-\frac{x^{6}}{6 !}\) Repeat part a using \(Y_{1}\) and \(Y_{3}=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}\) For what values of \(x\) does \(Y_{3}\) seem to be a good approximation for \(Y_{1} ?\) c. Use \(Y_{2}\) and \(Y_{3}\) to find approximations to the cosine function values below. Which function gives a better approximation? Is this what you expected? Explain. \(\begin{array}{llll}{\text { (1) } \cos -\frac{\pi}{6}} & {\text { (2) } \cos -\frac{\pi}{4}} & {\text { (3) } \cos -\pi}\end{array}\)

Find the amplitude of each function. \(y=\frac{1}{8} \sin x\)

City firefighters are told that they can use their 25 -foot long ladder provided the measure of the angle that the ladder makes with the ground is at least \(15^{\circ}\) and no more \(\operatorname{than} 75^{\circ}\) . a. If \(\theta\) represents the measure of the angle that the ladder makes with the ground in radians, what is a reasonable set of values for \(\theta ?\) Explain. b. Express as a function of \(\theta,\) the height \(h\) of the point at which the ladder will rest against a building. c. Graph the function from part b using the set of values for \(\theta\) from part a as the domain of the function. d. What is the highest point that the ladder is allowed to reach?

In \(3-14,\) sketch one cycle of the graph. $$ y=\tan x $$

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arctan (-1)\)

a. Sketch the graphs of \(y=\sin x\) and \(y=\csc x\) for \(-2 \pi \leq x \leq 2 \pi\) b. Name four values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which \(\sin x=\csc x\)

Find the period of each function. \(y=\sin x\)

In later courses, you will learn that the sine function can be written as the sum of an infinite sequence. In particular, for \(x\) in radians, the sine function can be approximated as the finite series: $$\sin x \approx x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}$$ a. Graph \(Y_{1}=\sin x\) and \(Y_{2}=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}\) on the graphing calculator. For what values of \(x\) does \(Y_{2}\) seem to be a good approximation for \(Y_{1} ?\) b. The next term of the sine approximation is \(-\frac{x^{7}}{7 !}\) . Repeat part a using \(Y_{1}\) and \(Y_{3}=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}\) . For what values of \(x\) does \(Y_{3}\) seem to be a good approximation for \(Y_{1} ?\) c. Use \(Y_{2}\) and \(Y_{3}\) to find approximations to the sine function values below. Which function gives a better approximation? Is this what you expected? Explain. (1) \(\sin \frac{\pi}{6}\) \(\qquad\) (2) \(\sin \frac{\pi}{4}\) \(\qquad\) \((3) \sin \pi\)

In \(3-14,\) sketch one cycle of the graph. $$ y=\tan \left(x-\frac{\pi}{2}\right) $$

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arcsin \left(-\frac{\sqrt{2}}{2}\right)\)

a. Sketch the graphs of \(y=\cos x\) and \(y=\sec x\) for \(-2 \pi \leq x \leq 2 \pi\) b. Name four values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which \(\cos x=\sec x\)

Find the period of each function. \(y=\cos x\)

a. Sketch the graphs of \(y=\tan x\) and \(y=\cot x\) for \(-\pi \leq x \leq \pi\) b. Name four values of \(x\) in the interval \(-\pi \leq x \leq \pi\) for which tan \(x=\cot x\)

Find the period of each function. \(y=\cos 3 x\)

In \(3-14,\) sketch one cycle of the graph. $$ y=-\cos x $$

List two values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which sec \(x\) is undefined.

Find the period of each function. \(y=\sin 2 x\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin 1\)

List two values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which csc \(x\) is undefined.

Motion that can be described by a sine or cosine function is called simple harmonic motion. During the day, a buoy in the ocean oscillates in simple harmonic motion. The frequency of the oscillation is equal to the reciprocal of the period. The distance between its high point and its low point is 1.5 meters. It takes the buoy 5 seconds to move between its low point and its high point, or 10 seconds for one complete oscillation from high point to high point. Let h(t) represent the height of the buoy as a function of time \(t .\) a. What is the amplitude of \(h(t) ?\) b. What is the period of \(h(t) ?\) c. What is the frequency of \(h(t) ?\) d. If \(h(0)\) represents the maximum height of the buoy, write an expression for h(t). e. Is there a value of \(t\) for which \(h(t)=1.5\) meters? Explain.

Find the period of each function. \(y=\cos \frac{1}{2} x\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos 1\)

List two values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which tan \(x\) is undefined.

Find the period of each function. \(y=\sin \frac{1}{3} x\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arctan 1\)

List two values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which cot \(x\) is undefined.

Find the period of each function. \(y=\sin 1.5 x\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin \left(\frac{\sqrt{3}}{2}\right)\)

The graphs of which two trigonometric functions have an asymptote at \(x=0 ?\)

Find the period of each function. \(y=\cos 0.75 x\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin \left(-\frac{\sqrt{3}}{2}\right)\)

The graphs of which two trigonometric functions have an asymptote at \(x=\frac{\pi}{2} ?\)

Find the phase shift of each function. \(y=\cos \left(x+\frac{\pi}{2}\right)\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos \frac{1}{2}\)

Using the graphs of each function, determine whether each function is even, odd, or neither. a. \(y=\tan x\) b. \(y=\csc x\) c. \(y=\sec x\) d. \(y=\cot x\)

Find the phase shift of each function. \(y=\cos \left(x-\frac{\pi}{2}\right)\)

a. On the same set of axes, sketch the graphs of \(y=2 \sin x\) and \(y=\cos x\) in the interval \(0 \leq x \leq 2 \pi\) b. How many points do the graphs of \(y=2 \sin x\) and \(y=\cos x\) have in common in the interval \(0 \leq x \leq 2 \pi ?\)

Find the phase shift of each function. \(y=\sin \left(x+\frac{\pi}{3}\right)\)

a. On the same set of axes, sketch the graphs of \(y=\tan x\) and \(y=\cos \left(x+\frac{\pi}{2}\right)\) in the interval \(-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}\) . b. How many points do the graphs of \(y=\tan x\) and \(y=\cos \left(x+\frac{\pi}{2}\right)\) have in common in the interval \(-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2} ?\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arctan \sqrt{3}\)

Find the phase shift of each function. \(y=\sin \left(x-\frac{\pi}{4}\right)\)

a. On the same set of axes, sketch the graphs of \(y=\sin 3 x\) and \(y=2 \cos 2 x\) in the interval \(0 \leq x \leq 2 \pi\) b. How many points do the graphs of \(y=\sin 3 x\) and \(y=2 \cos 2 x\) have in common in the interval \(0 \leq x \leq 2 \pi ?\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arctan (-\sqrt{3})\)

Find the phase shift of each function. \(y=\cos \left(x-\frac{\pi}{6}\right)\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos 0\)

Find the phase shift of each function. \(y=\sin 2\left(x+\frac{3 \pi}{4}\right)\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos 0\)

Find the phase shift of each function. \(y=\sin 2(x+\pi)\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin 0\)