9.5 Solving Trigonometric Equations

Find all possible exact solutions for the equation $\cos \theta = \frac{1}{2} .$

Find all possible exact solutions for the equation $\sin t = \frac{1}{2} .$

Solve the equation exactly: $2 \cos \theta - 3 = - 5 , 0 \leq \theta < 2 \pi .$

Solve exactly the following linear equation on the interval $\left[0 , 2 \pi\right) : 2 \sin x + 1 = 0.$

Solve the problem exactly: $2 \sin^{2} \theta - 1 = 0 , 0 \leq \theta < 2 \pi .$

Solve the following equation exactly: $\csc \theta = - 2 , 0 \leq \theta < 4 \pi .$

Solve the equation exactly: $\tan \left(\theta - \frac{\pi}{2}\right) = 1 , 0 \leq \theta < 2 \pi .$

Find all solutions for $\tan x = \sqrt{3} .$

Identify all exact solutions to the equation $2 \left(\tan x + 3\right) = 5 + \tan x , 0 \leq x < 2 \pi .$

Use a calculator to solve the equation $\sin \theta = 0.8 ,$ where $\theta$ is in radians.

Solve $\cos \theta = - 0.2.$

Solve the equation exactly: $\cos^{2} \theta + 3 \cos \theta - 1 = 0 , 0 \leq \theta < 2 \pi .$

Solve the equation exactly: $2 \sin^{2} \theta - 5 \sin \theta + 3 = 0 , 0 \leq \theta \leq 2 \pi .$

Solve $\sin^{2} \theta = 2 \cos \theta + 2 , 0 \leq \theta \leq 2 \pi .$ [Hint: Make a substitution to express the equation only in terms of cosine.]

Solve exactly:

Solve the equation quadratic in form exactly: $2 \sin^{2} \theta - 3 \sin \theta + 1 = 0 , 0 \leq \theta < 2 \pi .$

Solve the quadratic equation $2 \cos^{2} \theta + \cos \theta = 0.$

Use identities to solve exactly the trigonometric equation over the interval $0 \leq x < 2 \pi .$

Solve the equation exactly using a double-angle formula: $\cos \left(2 \theta\right) = \cos \theta .$

Solve the equation exactly using an identity: $3 \cos \theta + 3 = 2 \sin^{2} \theta , 0 \leq \theta < 2 \pi .$

Solve exactly: $\cos \left(2 x\right) = \frac{1}{2}$ on $\left[0 , 2 \pi\right) .$

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

Find all solutions exactly on the interval $0 \leq \theta < 2 \pi .$

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

Find all solutions exactly on the interval $0 \leq \theta < 2 \pi .$

$2 \cos \theta = − \sqrt{2}$

Find all solutions exactly on the interval $0 \leq \theta < 2 \pi .$

$\tan x = 1$

Find all solutions exactly on the interval $0 \leq \theta < 2 \pi .$

$4 \sin^{2} x - 2 = 0$

Solve exactly on $\left[0 , 2 \pi\right) .$

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

Solve exactly on $\left[0 , 2 \pi\right) .$

$2 \sin \theta = −1$

Solve exactly on $\left[0 , 2 \pi\right) .$

$2 \sin \left(3 \theta\right) = 1$

Solve exactly on $\left[0 , 2 \pi\right) .$

$2 \cos \left(3 \theta\right) = - \sqrt{2}$

Solve exactly on $\left[0 , 2 \pi\right) .$

$2 \sin \left(\pi \theta\right) = 1$

Find all exact solutions on $\left[0 , 2 \pi\right) .$

$\sec \left(x\right) \sin \left(x\right) - 2 \sin \left(x\right) = 0$

Find all exact solutions on $\left[0 , 2 \pi\right) .$

$2 \cos^{2} t + \cos \left(t\right) = 1$

Find all exact solutions on $\left[0 , 2 \pi\right) .$

$2 \sin \left(x\right) \cos \left(x\right) - \sin \left(x\right) + 2 \cos \left(x\right) - 1 = 0$

Find all exact solutions on $\left[0 , 2 \pi\right) .$

$\sec^{2} x = 1$

Find all exact solutions on $\left[0 , 2 \pi\right) .$

$8 \sin^{2} \left(x\right) + 6 \sin \left(x\right) + 1 = 0$

Solve with the methods shown in this section exactly on the interval $\left[0 , 2 \pi\right) .$

$\sin \left(3 x\right) \cos \left(6 x\right) - \cos \left(3 x\right) \sin \left(6 x\right) = −0.9$

Solve with the methods shown in this section exactly on the interval $\left[0 , 2 \pi\right) .$

$\cos \left(2 x\right) \cos x + \sin \left(2 x\right) \sin x = 1$

Solve with the methods shown in this section exactly on the interval $\left[0 , 2 \pi\right) .$

$9 \cos \left(2 \theta\right) = 9 \cos^{2} \theta - 4$

Solve with the methods shown in this section exactly on the interval $\left[0 , 2 \pi\right) .$

$\cos \left(2 t\right) = \sin t$

Solve exactly on the interval $\left[0 , 2 \pi\right) .$ Use the quadratic formula if the equations do not factor.

$\tan^{2} x - \sqrt{3} \tan x = 0$

Solve exactly on the interval $\left[0 , 2 \pi\right) .$ Use the quadratic formula if the equations do not factor.

$\sin^{2} x - 2 \sin x - 4 = 0$

Solve exactly on the interval $\left[0 , 2 \pi\right) .$ Use the quadratic formula if the equations do not factor.

$3 \cos^{2} x - 2 \cos x - 2 = 0$

Solve exactly on the interval $\left[0 , 2 \pi\right) .$ Use the quadratic formula if the equations do not factor.

$\tan^{2} x + 5 \tan x - 1 = 0$

Solve exactly on the interval $\left[0 , 2 \pi\right) .$ Use the quadratic formula if the equations do not factor.

$- \tan^{2} x - \tan x - 2 = 0$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$\sin^{2} x + \cos^{2} x = 0$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$\cos \left(2 x\right) - \cos x = 0$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$1 - \cos \left(2 x\right) = 1 + \cos \left(2 x\right)$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$10 \sin x \cos x = 6 \cos x$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$4 \cos^{2} x - 4 = 15 \cos x$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$8 \cos^{2} \theta = 3 - 2 \cos \theta$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$12 \sin^{2} t + \cos t - 6 = 0$

Find exact solutions on the interval $\left[0 , 2 \pi\right) .$ Look for opportunities to use trigonometric identities.

$\cos^{3} t = \cos t$

Algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros. Use an interval of $\left[0 , 2 \pi\right)$

$8 \cos^{2} x - 2 \cos x - 1 = 0$

Algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros. Use an interval of $\left[0 , 2 \pi\right)$

$2 \cos^{2} x - \cos x + 15 = 0$

Algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros. Use an interval of $\left[0 , 2 \pi\right)$

$2 \tan^{2} x + 7 \tan x + 6 = 0$

Use a calculator to find all solutions to four decimal places.

$\sin x = 0.27$

Use a calculator to find all solutions to four decimal places.

$\tan x = −0.34$

Solve the equations algebraically, and then use a calculator to find the values on the interval $\left[0 , 2 \pi\right) .$ Round to four decimal places.

$\tan^{2} x + 3 \tan x - 3 = 0$

Solve the equations algebraically, and then use a calculator to find the values on the interval $\left[0 , 2 \pi\right) .$ Round to four decimal places.

$\tan^{2} x - \sec x = 1$

Solve the equations algebraically, and then use a calculator to find the values on the interval $\left[0 , 2 \pi\right) .$ Round to four decimal places.

$2 \tan^{2} x + 9 \tan x - 6 = 0$

Find all solutions exactly to the equations on the interval $\left[0 , 2 \pi\right) .$

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

Find all solutions exactly to the equations on the interval $\left[0 , 2 \pi\right) .$

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

Find all solutions exactly to the equations on the interval $\left[0 , 2 \pi\right) .$

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

Find all solutions exactly to the equations on the interval $\left[0 , 2 \pi\right) .$

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

Find all solutions exactly to the equations on the interval $\left[0 , 2 \pi\right) .$

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

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 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?

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.)

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 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 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?

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?

Find all solutions exactly that exist on the interval $\left[0 , 2 \pi\right) .$

$\csc^{2} t = 3$

Find all solutions exactly that exist on the interval $\left[0 , 2 \pi\right) .$

$2 \sin \theta = −1$

Find all solutions exactly that exist on the interval $\left[0 , 2 \pi\right) .$

$9 \sin \omega - 2 = 4 \sin^{2} \omega$

Use basic identities to simplify the expression.

$\sec x \cos x + \cos x - \frac{1}{\sec x}$

Determine if the given identities are equivalent.

$\sin^{2} x + \sec^{2} x - 1 = \frac{\left(1 - \cos^{2} x\right) \left(1 + \cos^{2} x\right)}{\cos^{2} x}$

Find the exact value.

$\tan \left(\frac{7 \pi}{12}\right)$

Find the exact value.

$\sin \left(70°\right) \cos \left(25°\right) - \cos \left(70°\right) \sin \left(25°\right)$

Prove the identity.

$\cos \left(4 x\right) - \cos \left(3 x\right) \cos x = \sin^{2} x - 4 \cos^{2} x \sin^{2} x$

Simplify the expression.

$\frac{\tan \left(\frac{1}{2} x\right) + \tan \left(\frac{1}{8} x\right)}{1 - \tan \left(\frac{1}{8} x\right) \tan \left(\frac{1}{2} x\right)}$

Find the exact value.

$\tan \left(\sin^{- 1} \left(0\right) + \sin^{- 1} \left(\frac{1}{2}\right)\right)$

Find the exact value.

Find $\sin \left(2 \theta\right) , \cos \left(2 \theta\right) ,$ and $\tan \left(2 \theta\right)$ given $\sec \theta = - \frac{5}{3}$ and $\theta$ is in the interval $\left[\frac{\pi}{2} , \pi\right] .$

Find the exact value.

$\sec \left(\frac{3 \pi}{8}\right)$

$\sin \left(\frac{\beta}{2}\right) , \cos \left(\frac{\beta}{2}\right) , \tan \left(\frac{\beta}{2}\right) , \sin \left(\frac{\alpha}{2}\right) , \cos \left(\frac{\alpha}{2}\right) , \text{and}\textrm{ } \tan \left(\frac{\alpha}{2}\right)$

Prove the identity.

$\cot x \cos \left(2 x\right) = - \sin \left(2 x\right) + \cot x$

Rewrite the expression with no powers.

$\tan^{2} x \sin^{3} x$

Evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

$2 \sin \left(\frac{2 \pi}{3}\right) \sin \left(\frac{5 \pi}{6}\right)$

Evaluate the sum by using a product formula. Write the exact answer.

$\sin \left(\frac{\pi}{12}\right) - \sin \left(\frac{7 \pi}{12}\right)$

Change the functions from a product to a sum or a sum to a product.

$\sin \left(9 x\right) \cos \left(3 x\right)$

Change the functions from a product to a sum or a sum to a product.

$\sin \left(11 x\right) + \sin \left(2 x\right)$

Find all exact solutions on the interval $\left[0 , 2 \pi\right) .$

$\tan x + 1 = 0$

Find all exact solutions on the interval $\left[0 , 2 \pi\right) .$

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

Find all exact solutions on the interval $\left[0 , 2 \pi\right) .$

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

Find all exact solutions on the interval $\left[0 , 2 \pi\right) .$

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

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

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

Graph each side of the equation to find the approximate solutions on the interval $\left[0 , 2 \pi\right) .$

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

Simplify the given expression.

$\cos \left(− x\right) \sin x \cot x + \sin^{2} x$

Simplify the given expression.

$\csc \left(\theta\right) \cot \left(\theta\right) \left(\sec^{2} \theta - 1\right)$

Find the exact value.

$\cos \left(\frac{7 \pi}{12}\right)$

Find the exact value.

$\tan \left(\sin^{- 1} \left(\frac{\sqrt{2}}{2}\right) + \tan^{- 1} \sqrt{3}\right)$

Find the exact value.

$\cos \left(\frac{4 \pi}{3} + \theta\right)$

Simplify each expression. Do not evaluate.

$\cos^{2} \left(32°\right) \tan^{2} \left(32°\right)$

Find all exact solutions to the equation on $\left[0 , 2 \pi\right) .$

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

Find all exact solutions to the equation on $\left[0 , 2 \pi\right) .$

$\cos \left(2 x\right) + \sin^{2} x = 0$

Find all exact solutions to the equation on $\left[0 , 2 \pi\right) .$

Rewrite the expression as a product instead of a sum: $\cos \left(2 x\right) + \cos \left(- 8 x\right) .$

Rewrite the sum or difference as a product.

$2 \left(\sin \left(8 \theta\right) - \sin \left(4 \theta\right)\right)$

Rewrite the sum or difference as a product.

Find the solutions of $\sec^{2} x - 2 \sec x = 15$ on the interval $\left[0 , 2 \pi\right)$ algebraically; then graph both sides of the equation to determine the answer.

Find all solutions exactly on the interval $0 \leq \theta \leq \pi$

$\sqrt{3} \cot \left(y\right) = 1$

Find all solutions exactly on the interval $0 \leq \theta \leq \pi$

Find $\sin \left(\frac{\theta}{2}\right) , \cos \left(\frac{\theta}{2}\right) ,$ and $\tan \left(\frac{\theta}{2}\right)$ given $\cos \theta = \frac{7}{25}$ and $\theta$ is in quadrant IV.

Prove the identity.

$\tan^{3} x - \tan x \sec^{2} x = \tan \left(- x\right)$

Prove the identity.

$\frac{\sin \left(2 x\right)}{\sin x} - \frac{\cos \left(2 x\right)}{\cos x} = \sec x$

Prove the identity.

The displacement $h \left(t\right)$ in centimeters of a mass suspended by a spring is modeled by the function $h \left(t\right) = \frac{1}{4} \sin \left(120 \pi t\right) ,$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Prove the identity.

Two frequencies of sound are played on an instrument governed by the equation $n \left(t\right) = 8 \cos \left(20 \pi t\right) \cos \left(1000 \pi t\right) .$ What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

Prove the identity.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring $t$ seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.