# 9.5 Solving trigonometric equations  (Page 8/10)

 Page 8 / 10

$6\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+13\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-6$

${\mathrm{tan}}^{2}x-\mathrm{sec}\text{\hspace{0.17em}}x=1$

$1.0472,3.1416,5.2360$

${\mathrm{sin}}^{2}x-2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x=0$

$2\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+9\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-6=0$

$0.5326,1.7648,3.6742,4.9064$

$4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+\mathrm{sin}\left(2x\right)\mathrm{sec}\text{\hspace{0.17em}}x-3=0$

## Extensions

For the following exercises, find all solutions exactly to the equations on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).$

${\mathrm{csc}}^{2}x-3\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x-4=0$

${\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\frac{3\pi }{2}$

${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x-1=0$

${\mathrm{sin}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)+{\mathrm{cos}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)=0$

$\frac{\pi }{2},\frac{3\pi }{2}$

$3\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x+2+{\mathrm{sin}}^{2}x-{\mathrm{tan}}^{2}x+{\mathrm{cos}}^{2}x=0$

${\mathrm{sin}}^{2}x-1+2\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)-{\mathrm{cos}}^{2}x=1$

There are no solutions.

${\mathrm{tan}}^{2}x-1-{\mathrm{sec}}^{3}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=0$

$\frac{\mathrm{sin}\left(2x\right)}{{\mathrm{sec}}^{2}x}=0$

$0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$

$\frac{\mathrm{sin}\left(2x\right)}{2{\mathrm{csc}}^{2}x}=0$

$2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-\mathrm{cos}\text{\hspace{0.17em}}x-5=0$

There are no solutions.

$\frac{1}{{\mathrm{sec}}^{2}x}+2+{\mathrm{sin}}^{2}x+4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x=4$

## Real-world applications

An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?

${7.2}^{\circ }$

If a loading ramp is placed next to a truck, at a height of 4 feet, and the ramp is 15 feet long, what angle does the ramp make with the ground?

If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?

${5.7}^{\circ }$

A woman is watching a launched rocket currently 11 miles in altitude. If she is standing 4 miles from the launch pad, at what angle is she looking up from horizontal?

An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is she looking down at him from horizontal? (Hint: this is called the angle of depression.)

${82.4}^{\circ }$

A woman is standing 8 meters away from a 10-meter tall building. At what angle is she looking to the top of the building?

A man is standing 10 meters away from a 6-meter tall building. Someone at the top of the building is looking down at him. At what angle is the person looking at him?

${31.0}^{\circ }$

A 20-foot tall building has a shadow that is 55 feet long. What is the angle of elevation of the sun?

A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?

${88.7}^{\circ }$

A spotlight on the ground 3 meters from a 2-meter tall man casts a 6 meter shadow on a wall 6 meters from the man. At what angle is the light?

A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?

${59.0}^{\circ }$

For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.

A person does a handstand with his feet touching a wall and his hands 1.5 feet away from the wall. If the person is 6 feet tall, what angle do his feet make with the wall?

A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?

${36.9}^{\circ }$

A 23-foot ladder is positioned next to a house. If the ladder slips at 7 feet from the house when there is not enough traction, what angle should the ladder make with the ground to avoid slipping?

## Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).$

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey