삼각함수를 포함한 방정식과 부등식

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

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

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

$3\,{\sec }^{2}x+2+{\sin }^{2}x-{\tan }^{2}x+{\cos }^{2}x=0$

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

${\sin }^{2}x-1+2\,\cos (2x)-{\cos }^{2}x=1$

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

${\tan }^{2}x-1-{\sec }^{3}x\,\cos \,x=0$

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

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

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

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

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

$2\,{\cos }^{2}x-{\sin }^{2}x-\cos \,x-5=0$

For the following exercises, find all solutions exactly to the equations on the interval $[0,2\pi ).$

$\frac{1}{{\sec }^{2}x}+2+{\sin }^{2}x+4\,{\cos }^{2}x=4$

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?

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?

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 the astronaut looking down at him from horizontal? (Hint: this is called the angle of depression.)

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?

Issa is standing 10 meters away from a 6-meter tall building. Travis is at the top of the building looking down at Issa. At what angle is Travis looking at Issa?

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?

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?

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