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Algebra 2 and Trigonometry

Ann Xavier Gantert

Chapter 11

GRAPHS OF TRIGONOMETRIC FUNCTIONS

Educators


Problem 1

Is the graph of $y=\sin x$ symmetric with respect to a reflection in the origin? Justify your answer.

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Problem 2

Is the graph of $y=\sin x$ symmetric with respect to the translation $T_{-2 \pi, 0} ?$ Justify your answer.

Martha R.
Numerade Educator

Problem 3

Sketch the graph of $y=\sin x$ in the interval $0 \leq x \leq 4 \pi$
a. In the interval $0 \leq x \leq 4 \pi,$ for what values of $x$ is the graph of $y=\sin x$ increasing?
b. In the interval $0 \leq x \leq 4 \pi$ , for what values of $x$ is the graph of $y=\sin x$ decreasing?
c. How many cycles of the graph of $y=\sin x$ are in the interval $0 \leq x \leq 4 \pi ?$

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Problem 4

What is the maximum value of $y$ on the graph of $y=\sin x ?$

Martha R.
Numerade Educator

Problem 5

What is the minimum value of $y$ on the graph of $y=\sin x ?$

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Problem 6

What is the period of the sine function?

Martha R.
Numerade Educator

Problem 7

Is the sine function one-to-one? Justify your answer.

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Problem 8

a. Point $P$ is a point on the unit circle. The $y$ -coordinate of $P$ is $\sin \frac{\pi}{3} .$ What is the $x$-coordinate of $P ?$
b. Point $A$ is a point on the graph $y=\sin x .$ The $y$ -coordinate of $A$ is $\sin \frac{\pi}{3} .$ What is the $x$-coordinate of $A ?$

Martha R.
Numerade Educator

Problem 9

A function $\mathrm{f}$ is odd if and only if $\mathrm{f}(x)=-\mathrm{f}(-x)$ for all $x$ in the domain of the function. Note that a function is odd if it is symmetric with respect to the origin. In other words, the function is its own image under a reflection about the origin.
a. Draw a unit circle and any first-quadrant angle $R O P$ in standard position, with point $P$ on the unit circle. Let $\mathrm{m} \angle R O P=\theta .$
b. On the same set of axes, draw an angle in standard position with measure $-\theta .$ What is the relationship between $\theta$ and $-\theta ?$ Between $\sin \theta$ and $\sin (-\theta) ?$
c. Repeat steps a and b for second- third-, and fourth-quadrant angles. Does $\sin \theta=\sin$ $(-\theta)$ for second-, third-, and fourth-quadrant angles? Justify your answer.
d. Does $\sin \theta=-\sin (-\theta)$ for quadrantal angles? Explain.
e. Do parts a-d show that $y=\sin x$ is an odd function? Justify your answer.

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Problem 10

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?

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Problem 11

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) $\sin \frac{\pi}{4}$ $\qquad$ $(3) \sin \pi$

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