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For the following exercises, find all solutions exactly on the interval $\text{\hspace{0.17em}}0\le \theta <2\pi .$
$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{-}\sqrt{2}$
$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\sqrt{3}$
$\frac{\pi}{3},\frac{2\pi}{3}$
$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =1$
$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\mathrm{-}\sqrt{2}$
$\frac{3\pi}{4},\frac{5\pi}{4}$
$\mathrm{tan}\text{\hspace{0.17em}}\theta =\mathrm{-1}$
$\mathrm{tan}\text{\hspace{0.17em}}x=1$
$\frac{\pi}{4},\frac{5\pi}{4}$
$\mathrm{cot}\text{\hspace{0.17em}}x+1=0$
$4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-2=0$
$\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}$
${\mathrm{csc}}^{2}x-4=0$
For the following exercises, solve exactly on $\text{\hspace{0.17em}}[0,2\pi ).$
$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\sqrt{2}$
$\frac{\pi}{4},\frac{7\pi}{4}$
$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\mathrm{-1}$
$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{-1}$
$\frac{7\pi}{6},\frac{11\pi}{6}$
$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{-}\sqrt{3}$
$2\text{\hspace{0.17em}}\mathrm{sin}\left(3\theta \right)=1$
$\frac{\pi}{18},\frac{5\pi}{18},\frac{13\pi}{18},\frac{17\pi}{18},\frac{25\pi}{18},\frac{29\pi}{18}$
$2\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right)=\sqrt{3}$
$2\text{\hspace{0.17em}}\mathrm{cos}\left(3\theta \right)=-\sqrt{2}$
$\frac{3\pi}{12},\frac{5\pi}{12},\frac{11\pi}{12},\frac{13\pi}{12},\frac{19\pi}{12},\frac{21\pi}{12}$
$\mathrm{cos}\left(2\theta \right)=-\frac{\sqrt{3}}{2}$
$2\text{\hspace{0.17em}}\mathrm{sin}\left(\pi \theta \right)=1$
$\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}$
$2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi}{5}\theta \right)=\sqrt{3}$
For the following exercises, find all exact solutions on $\text{\hspace{0.17em}}\left[0,2\pi \right).$
$\mathrm{sec}(x)\mathrm{sin}(x)-2\text{\hspace{0.17em}}\mathrm{sin}(x)=0$
$0,\frac{\pi}{3},\pi ,\frac{5\pi}{3}$
$\mathrm{tan}(x)-2\text{\hspace{0.17em}}\mathrm{sin}(x)\mathrm{tan}(x)=0$
$2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+\mathrm{cos}\left(t\right)=1$
$\frac{\pi}{3},\pi ,\frac{5\pi}{3}$
$2\text{\hspace{0.17em}}{\mathrm{tan}}^{2}(t)=3\text{\hspace{0.17em}}\mathrm{sec}(t)$
$2\text{\hspace{0.17em}}\mathrm{sin}(x)\mathrm{cos}(x)-\mathrm{sin}(x)+2\text{\hspace{0.17em}}\mathrm{cos}(x)-1=0$
$\frac{\pi}{3},\frac{3\pi}{2},\frac{5\pi}{3}$
${\mathrm{cos}}^{2}\theta =\frac{1}{2}$
${\mathrm{tan}}^{2}\left(x\right)=-1+2\text{\hspace{0.17em}}\mathrm{tan}\left(-x\right)$
$8\text{\hspace{0.17em}}{\mathrm{sin}}^{2}(x)+6\text{\hspace{0.17em}}\mathrm{sin}(x)+1=0$
$\pi -{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right),\frac{7\pi}{6},\frac{11\pi}{6},2\pi +{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right)$
${\mathrm{tan}}^{5}(x)=\mathrm{tan}(x)$
For the following exercises, solve with the methods shown in this section exactly on the interval $\text{\hspace{0.17em}}[0,2\pi ).$
$\mathrm{sin}(3x)\mathrm{cos}(6x)-\mathrm{cos}(3x)\mathrm{sin}(6x)=\mathrm{-0.9}$
$$\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{\pi}{3}-\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{2\pi}{3}+\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\pi -\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{4\pi}{3}+\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right),\frac{5\pi}{3}-\frac{1}{3}\left({\mathrm{sin}}^{-1}\left(\frac{9}{10}\right)\right)$$
$\mathrm{sin}(6x)\mathrm{cos}(11x)-\mathrm{cos}(6x)\mathrm{sin}(11x)=\mathrm{-0.1}$
$\mathrm{cos}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x+\mathrm{sin}\left(2x\right)\mathrm{sin}\text{\hspace{0.17em}}x=1$
$0$
$6\text{\hspace{0.17em}}\mathrm{sin}\left(2t\right)+9\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=0$
$9\text{\hspace{0.17em}}\mathrm{cos}\left(2\theta \right)=9\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\theta -4$
$\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}$
$\mathrm{sin}\left(2t\right)=\mathrm{cos}\text{\hspace{0.17em}}t$
$\mathrm{cos}\left(2t\right)=\mathrm{sin}\text{\hspace{0.17em}}t$
$\frac{3\pi}{2},\frac{\pi}{6},\frac{5\pi}{6}$
$\mathrm{cos}(6x)-\mathrm{cos}(3x)=0$
For the following exercises, solve exactly on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).\text{\hspace{0.17em}}$ Use the quadratic formula if the equations do not factor.
${\mathrm{tan}}^{2}x-\sqrt{3}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=0$
$0,\frac{\pi}{3},\pi ,\frac{4\pi}{3}$
${\mathrm{sin}}^{2}x+\mathrm{sin}\text{\hspace{0.17em}}x-2=0$
${\mathrm{sin}}^{2}x-2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-4=0$
There are no solutions.
$5\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x+3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x-1=0$
$3\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x-2=0$
${\mathrm{cos}}^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right),2\pi -{\mathrm{cos}}^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right)$
$5\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-1=0$
${\mathrm{tan}}^{2}x+5\mathrm{tan}\text{\hspace{0.17em}}x-1=0$
${\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right),\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),2\pi +{\mathrm{tan}}^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right)$
${\mathrm{cot}}^{2}x=-\mathrm{cot}\text{\hspace{0.17em}}x$
$-{\mathrm{tan}}^{2}x-\mathrm{tan}\text{\hspace{0.17em}}x-2=0$
There are no solutions.
For the following exercises, find exact solutions on the interval $\text{\hspace{0.17em}}[0,2\pi ).\text{\hspace{0.17em}}$ Look for opportunities to use trigonometric identities.
${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x-\mathrm{sin}\text{\hspace{0.17em}}x=0$
${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=0$
There are no solutions.
$\mathrm{sin}\left(2x\right)-\mathrm{sin}\text{\hspace{0.17em}}x=0$
$\mathrm{cos}\left(2x\right)-\mathrm{cos}\text{\hspace{0.17em}}x=0$
$0,\frac{2\pi}{3},\frac{4\pi}{3}$
$\frac{2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x}{2-{\mathrm{sec}}^{2}x}-{\mathrm{sin}}^{2}x={\mathrm{cos}}^{2}x$
$1-\mathrm{cos}(2x)=1+\mathrm{cos}(2x)$
$\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}$
${\mathrm{sec}}^{2}x=7$
$10\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=6\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x$
${\mathrm{sin}}^{-1}\left(\frac{3}{5}\right),\frac{\pi}{2},\pi -{\mathrm{sin}}^{-1}\left(\frac{3}{5}\right),\frac{3\pi}{2}$
$\mathrm{-3}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=15\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t$
$4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-4=15\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x$
${\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right)$
$8\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+6\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+1=0$
$8\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\theta =3-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta $
$\frac{\pi}{3},{\mathrm{cos}}^{-1}\left(-\frac{3}{4}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{3}{4}\right),\frac{5\pi}{3}$
$6\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x+7\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-8=0$
$12\text{\hspace{0.17em}}{\mathrm{sin}}^{2}t+\mathrm{cos}\text{\hspace{0.17em}}t-6=0$
${\mathrm{cos}}^{-1}\left(\frac{3}{4}\right),{\mathrm{cos}}^{-1}\left(-\frac{2}{3}\right),2\pi -{\mathrm{cos}}^{-1}\left(-\frac{2}{3}\right),2\pi -{\mathrm{cos}}^{-1}\left(\frac{3}{4}\right)$
$\mathrm{tan}\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x$
${\mathrm{cos}}^{3}t=\mathrm{cos}\text{\hspace{0.17em}}t$
$0,\frac{\pi}{2},\pi ,\frac{3\pi}{2}$
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.
$6\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-5\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+1=0$
$8\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x-1=0$
$\frac{\pi}{3},{\mathrm{cos}}^{\mathrm{-1}}\left(-\frac{1}{4}\right),2\pi -{\mathrm{cos}}^{\mathrm{-1}}\left(-\frac{1}{4}\right),\frac{5\pi}{3}$
$100\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+20\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-3=0$
$2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-\mathrm{cos}\text{\hspace{0.17em}}x+15=0$
There are no solutions.
$20\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-27\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+7=0$
$2\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+7\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x+6=0$
$\pi +{\mathrm{tan}}^{\mathrm{-1}}\left(\mathrm{-2}\right),\pi +{\mathrm{tan}}^{\mathrm{-1}}\left(-\frac{3}{2}\right),2\pi +{\mathrm{tan}}^{\mathrm{-1}}\left(\mathrm{-2}\right),2\pi +{\mathrm{tan}}^{\mathrm{-1}}\left(-\frac{3}{2}\right)$
$130\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+69\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-130=0$
For the following exercises, use a calculator to find all solutions to four decimal places.
$\mathrm{sin}\text{\hspace{0.17em}}x=0.27$
$2\pi k+0.2734,2\pi k+2.8682$
$\mathrm{sin}\text{\hspace{0.17em}}x=\mathrm{-0.55}$
$\mathrm{tan}\text{\hspace{0.17em}}x=\mathrm{-0.34}$
$\pi k-0.3277$
$\mathrm{cos}\text{\hspace{0.17em}}x=0.71$
For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $\text{\hspace{0.17em}}[0,2\pi ).\text{\hspace{0.17em}}$ Round to four decimal places.
${\mathrm{tan}}^{2}x+3\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-3=0$
$0.6694,1.8287,3.8110,4.9703$
$6\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+13\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=\mathrm{-6}$
${\mathrm{tan}}^{2}x-\mathrm{sec}\text{\hspace{0.17em}}x=1$
$1.0472,3.1416,5.2360$
${\mathrm{sin}}^{2}x-2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x=0$
$2\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+9\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-6=0$
$0.5326,1.7648,3.6742,4.9064$
$4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+\mathrm{sin}\left(2x\right)\mathrm{sec}\text{\hspace{0.17em}}x-3=0$
For the following exercises, find all solutions exactly to the equations on the interval $\text{\hspace{0.17em}}[0,2\pi ).$
${\mathrm{csc}}^{2}x-3\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x-4=0$
${\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\frac{3\pi}{2}$
${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x-1=0$
${\mathrm{sin}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)+{\mathrm{cos}}^{2}x\left(1-{\mathrm{sin}}^{2}x\right)=0$
$\frac{\pi}{2},\frac{3\pi}{2}$
$3\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x+2+{\mathrm{sin}}^{2}x-{\mathrm{tan}}^{2}x+{\mathrm{cos}}^{2}x=0$
${\mathrm{sin}}^{2}x-1+2\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)-{\mathrm{cos}}^{2}x=1$
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.
$\frac{1}{{\mathrm{sec}}^{2}x}+2+{\mathrm{sin}}^{2}x+4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x=4$
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?
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