# 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 $\text{\hspace{0.17em}}\left[0,2\pi \right).$

#### Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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