Answer to Question #283436 in Trigonometry for kib888

Question

Find the length of an arc in a circle of radius 10 centimeters subtended by the central angle of 50°. Show your work.

2. Graph on [-4π, 4π] and verbalize how the graph varies from the graphs of . 

Graph  on the window [−5π, 5π] and describe freely what the graph shows. You can use www.desmos.com/calculator to obtain the graphs.

3. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building? Show the steps of your reasoning.  

Answer 1

1. The circumference of the circle of radius $r$ is $L=2\pi r.$

An arc is a portion of the outline of a circle.

The length of the arc of a circle of radius $r$ that subtends an angle of $θ$

 in degrees at the center is

$$Arc\ length=2\pi r(\dfrac{\theta}{360\degree})$$

Given $r=10\ cm, \theta=50\degree.$

$$Arc\ length=2\pi (10)(\dfrac{50\degree}{360\degree})=\dfrac{25\pi}{9}(cm)$$

$$=\dfrac{25\pi}{9}(cm)\approx8.7266(cm)$$

2.

(i)

$$f(x)=x\sin x$$

First draw the graph of $f(x)=±x.$

The function $f(x)=x\sin x$ is even. The graph is symmetric with respect to the $y$ -axis.

Points $(-\dfrac{\pi}{2}+2\pi n,\dfrac{\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=-x$

Points $(\dfrac{\pi}{2}+2\pi n,\dfrac{\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=x$

Points $(-\dfrac{3\pi}{2}+2\pi n,-\dfrac{3\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=x$

Points $(\dfrac{3\pi}{2}+2\pi n,-\dfrac{3\pi}{2}+2\pi n),n\in \Z$ lie on the graph of $f(x)=-x$

(ii)

The function $f(x)=\dfrac{\sin x}{x}$ is not defined at $x=0.$

$$\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$$

The function $f(x)=\dfrac{\sin x}{x}$ has a removable discontinuity at $x=0.$

The function $f(x)=\dfrac{\sin x}{x}$ is even. The graph is symmetric with respect to the $y$ -axis.

$y\to0$ as $x\to\pm \infin.$

The graph of $f(x)=\dfrac{\sin x}{x}$ is decaying oscillations (oscillations of continuously decreasing amplitude). The oscillations never stop, but go on decreasing in strength. The amplitude of oscillations at any point $x\not=0$ is $1/x.$

3.

From right triangle

$$h=L\sin \theta$$

$$h=23\sin 80\degree \ ft\approx22.65\ ft$$