# 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.

#### Questions & Answers

can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
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
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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