# 5.4 Right triangle trigonometry  (Page 7/12)

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A 23-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}80°.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building?

22.6506 ft

The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

368.7633 ft

Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be $\text{\hspace{0.17em}}60°,$ how far from the base of the tree am I?

## Angles

For the following exercises, convert the angle measures to degrees.

$45°$

$-\frac{5\pi }{3}$

For the following exercises, convert the angle measures to radians.

-210°

$-\frac{7\pi }{6}$

180°

Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85°.

10.385 meters

Find the area of the sector of a circle with diameter 32 feet and an angle of $\text{\hspace{0.17em}}\frac{3\pi }{5}\text{\hspace{0.17em}}$ radians.

For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.

$420°$

$60°$

$-80°$

For the following exercises, find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with the given angle.

$-\text{\hspace{0.17em}}\frac{20\pi }{11}$

$\frac{2\pi }{11}$

$\frac{14\pi }{5}$

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

-210° 75°

$\frac{5\pi }{4}$ $-\frac{\pi }{3}$

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.

1036.73 miles per hour

A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour?

## Unit Circle: Sine and Cosine Functions

Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{3}.$

$\frac{\sqrt{3}}{2}$

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{4}.$

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\pi .$

–1

State the reference angle for $\text{\hspace{0.17em}}300°.$

State the reference angle for $\text{\hspace{0.17em}}\frac{3\pi }{4}.$

$\frac{\pi }{4}$

Compute cosine of $\text{\hspace{0.17em}}330°.$

Compute sine of $\text{\hspace{0.17em}}\frac{5\pi }{4}.$

$-\frac{\sqrt{2}}{2}$

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

$\left[–1,1\right]$

## The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{4}$

1

$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\sqrt{2}$

For the following exercises, use reference angles to evaluate the given expression.

$\mathrm{sec}\text{\hspace{0.17em}}\frac{11\pi }{3}$

$\mathrm{sec}\text{\hspace{0.17em}}315°$

$\sqrt{2}$

If $\text{\hspace{0.17em}}\mathrm{sec}\left(t\right)=-2.5\text{\hspace{0.17em}}$ , what is the $\text{\hspace{0.17em}}\text{sec}\left(-t\right)?$

If $\text{\hspace{0.17em}}\text{tan}\left(t\right)=-0.6,$ what is the $\text{\hspace{0.17em}}\text{tan}\left(-t\right)?$

0.6

If $\text{\hspace{0.17em}}\text{tan}\left(t\right)=\frac{1}{3},$ find $\text{\hspace{0.17em}}\text{tan}\left(t-\pi \right).$

If $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{\sqrt{2}}{2},$ find $\text{\hspace{0.17em}}\text{sin}\left(t+2\pi \right).$

$\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}-\frac{\sqrt{2}}{2}$

Which trigonometric functions are even?

Which trigonometric functions are odd?

sine, cosecant, tangent, cotangent

## Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{\sqrt{3}}{3}$

$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{cos}\left(\frac{\pi }{2}\right)=\mathrm{sin}\left(\text{__°}\right)$

0

$\mathrm{csc}\left(18\text{°}\right)=\mathrm{sec}\left(\text{__°}\right)$

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

$\mathrm{cos}\text{\hspace{0.17em}}B=\frac{3}{5},a=6$

$b=8,c=10$

$\mathrm{tan}\text{\hspace{0.17em}}A=\frac{5}{9},b=6$

For the following exercises, use [link] to evaluate each trigonometric function.

$\mathrm{sin}\text{\hspace{0.17em}}A$

$\frac{11\sqrt{157}}{157}$

$\mathrm{tan}\text{\hspace{0.17em}}B$

For the following exercises, solve for the unknown sides of the given triangle.  A 15-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}70°.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building?

14.0954 ft

The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

## Practice test

Convert $\text{\hspace{0.17em}}\frac{5\pi }{6}\text{\hspace{0.17em}}$ radians to degrees.

$150°$

Convert $\text{\hspace{0.17em}}-620°\text{\hspace{0.17em}}$ to radians.

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of $\text{\hspace{0.17em}}30°.$

6.283 centimeters

Find the area of the sector with radius of 8 feet and an angle of $\text{\hspace{0.17em}}\frac{5\pi }{4}\text{\hspace{0.17em}}$ radians.

Find the angle between $\text{\hspace{0.17em}}0°\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{360°}\text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}375°.$

$15°$

Find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with $\text{\hspace{0.17em}}-\frac{4\pi }{7}.$

Draw the angle $\text{\hspace{0.17em}}315°\text{\hspace{0.17em}}$ in standard position on the Cartesian plane. Draw the angle $\text{\hspace{0.17em}}-\frac{\pi }{6}\text{\hspace{0.17em}}$ in standard position on the Cartesian plane.

A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

3.351 feet per second, $\text{\hspace{0.17em}}\frac{2\pi }{75}\text{\hspace{0.17em}}$ radians per second

Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{6}.$

Compute sine of $\text{\hspace{0.17em}}240°.$

$-\frac{\sqrt{3}}{2}$

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

$\left[–1,1\right]$

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{4}.$

Find the exact value of $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{3}.$

$\sqrt{3}$

Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}\frac{7\pi }{4}.$

Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}210°.$

$\frac{\sqrt{3}}{3}$

If $\text{\hspace{0.17em}}\text{csc}\text{\hspace{0.17em}}t=0.68,$ what is the $\text{\hspace{0.17em}}\text{csc}\left(-t\right)?$

If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}\text{t}=\frac{\sqrt{3}}{2},$ find $\text{\hspace{0.17em}}\text{cos}\left(t-2\pi \right).$

$\frac{\sqrt{3}}{2}$

Which trigonometric functions are even?

Find the missing angle: $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}\right)=\mathrm{sin}\left(___\right)$

$\frac{\pi }{3}$

Find the missing sides of the triangle $\text{\hspace{0.17em}}ABC:\mathrm{sin}\text{\hspace{0.17em}}B=\frac{3}{4},c=12$

Find the missing sides of the triangle. $a=\frac{9}{2},b=\frac{9\sqrt{3}}{2}$

The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert By By By  By   By By   