삼각함수를 포함한 방정식과 부등식

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 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?

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 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?

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

$\tan \,x+1=0$

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

$2\,\sin (2x)+\sqrt{2}=0$

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

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

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

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

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

$2\,{\sin }^{2}x+5\,\sin \,x+3=0$

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

$\cos \,x-5\,\sin (2x)=0$

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

$\frac{1}{{\sec }^{2}x}+2+{\sin }^{2}x+4\,{\cos }^{2}x=0$

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $[0,2\pi ).$ Round to four decimal places.

$\sqrt{3}\,{\cot }^{2}x+\cot \,x=1$

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $[0,2\pi ).$ Round to four decimal places.

${\csc }^{2}x-3\,\csc \,x-4=0$

For the following exercises, graph each side of the equation to find the approximate solutions on the interval $[0,2\pi ).$

$20\,{\cos }^{2}x+21\,\cos \,x+1=0$

For the following exercises, graph each side of the equation to find the approximate solutions on the interval $[0,2\pi ).$

${\sec }^{2}x-2\,\sec \,x=15$

Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.

When solving a trigonometric equation involving more than one trig function, do we always want to try to rewrite the equation so it is expressed in terms of one trigonometric function? Why or why not?

When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?

For the following exercises, find all solutions exactly on the interval $0\le \theta <2\pi .$

$2\sin \,\theta =-\sqrt{2}$

For the following exercises, find all solutions exactly on the interval $0\le \theta <2\pi .$

$2\sin \,\theta =\sqrt{3}$

For the following exercises, find all solutions exactly on the interval $0\le \theta <2\pi .$

$2\cos \,\theta =1$

For the following exercises, find all solutions exactly on the interval $0\le \theta <2\pi .$

$2\cos \,\theta =-\sqrt{2}$