# 9.5 Solving trigonometric equations  (Page 8/10)

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

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

$1.0472,3.1416,5.2360$

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

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

$0.5326,1.7648,3.6742,4.9064$

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

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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
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Ajith
exponential series
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what is subgroup
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e power cos hyperbolic (x+iy)
10y
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prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
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why is {2kπ} union {kπ}={kπ}? when k belong to integer
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if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
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Yes
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Thank you
Dua
give me treganamentry question
Solve 2cos x + 3sin x = 0.5