Answer to Question #344941 in Trigonometry for Kelvin

Question #344941

Written Assignment unit 7




Complete the following questions utilizing the concepts introduced in this unit.



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





2. Graph f(x) = x Sin x on [-4π, 4π] and verbalize how the graph varies from the graphs of f(x) + or - x.



Graph f(x) =Sin x / x 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.




1
Expert's answer
2022-05-31T13:47:16-0400

1. The circumference of the circle is "L=2\\pi r=2\\pi(10)=20\\pi(cm)"

The length "l" of yhe arc in a circle subtended by the central angle of 50° is


"l=L(\\dfrac{50\\degree}{360\\degree})=\\dfrac{20\\pi(5)}{36}=\\dfrac{25\\pi}{9}(cm)"

2.

First draw the graph of "f(x)=\u00b1x."

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"

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