<< Chapter < Page Chapter >> Page >

Find all solutions for tan x = 3 .

π 3 ± π k

Got questions? Get instant answers now!

Identify all solutions to the equation involving tangent

Identify all exact solutions to the equation 2 ( tan x + 3 ) = 5 + tan x , 0 x < 2 π .

We can solve this equation using only algebra. Isolate the expression tan x on the left side of the equals sign.

2 ( tan x ) + 2 ( 3 ) = 5 + tan x 2 tan x + 6 = 5 + tan x 2 tan x tan x = 5 6 tan x = 1

There are two angles on the unit circle that have a tangent value of −1 : θ = 3 π 4 and θ = 7 π 4 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve trigonometric equations using a calculator

Not all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle other than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either degrees or radians, depending on the criteria of the given problem.

Using a calculator to solve a trigonometric equation involving sine

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

Make sure mode is set to radians. To find θ , use the inverse sine function. On most calculators, you will need to push the 2 ND button and then the SIN button to bring up the sin 1 function. What is shown on the screen is sin 1 ( . The calculator is ready for the input within the parentheses. For this problem, we enter sin 1 ( 0.8 ) , and press ENTER. Thus, to four decimals places,

sin 1 ( 0.8 ) 0.9273

The solution is

0.9273 ± 2 π k

The angle measurement in degrees is

θ 53.1 θ 180 53.1    126.9
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using a calculator to solve a trigonometric equation involving secant

Use a calculator to solve the equation sec θ = −4 , giving your answer in radians.

We can begin with some algebra.

sec θ = 4 1 cos θ = 4 cos θ = 1 4

Check that the MODE is in radians. Now use the inverse cosine function.

cos 1 ( 1 4 ) 1.8235                   θ 1.8235 + 2 π k

Since π 2 1.57 and π 3.14 , 1.8235 is between these two numbers, thus θ 1 .8235 is in quadrant II. Cosine is also negative in quadrant III. Note that a calculator will only return an angle in quadrants I or II for the cosine function, since that is the range of the inverse cosine. See [link] .

Graph of angles theta =approx 1.8235, theta prime =approx pi - 1.8235 = approx 1.3181, and then theta prime = pi + 1.3181 = approx 4.4597

So, we also need to find the measure of the angle in quadrant III. In quadrant III, the reference angle is θ ' π 1 .8235 1 .3181 . The other solution in quadrant III is π + 1 .3181 4 .4597 .

The solutions are 1.8235 ± 2 π k and 4.4597 ± 2 π k .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve cos θ = 0.2.

θ 1.7722 ± 2 π k and θ 4.5110 ± 2 π k

Got questions? Get instant answers now!

Solving trigonometric equations in quadratic form

Solving a quadratic equation may be more complicated, but once again, we can use algebra as we would for any quadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation, or is there only one? Which trigonometric function is squared? If there is only one function represented and one of the terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such as x or u . If substitution makes the equation look like a quadratic equation, then we can use the same methods for solving quadratics to solve the trigonometric equations.

Solving a trigonometric equation in quadratic form

Solve the equation exactly: cos 2 θ + 3 cos θ 1 = 0 , 0 θ < 2 π .

We begin by using substitution and replacing cos θ with x . It is not necessary to use substitution, but it may make the problem easier to solve visually. Let cos θ = x . We have

x 2 + 3 x 1 = 0

The equation cannot be factored, so we will use the quadratic formula x = b ± b 2 4 a c 2 a .

x = 3 ± ( 3 ) 2 4 ( 1 ) ( 1 ) 2    = 3 ± 13 2

Replace x with cos θ , and solve. Thus,

cos θ = 3 ± 13 2       θ = cos 1 ( 3 + 13 2 )

Note that only the + sign is used. This is because we get an error when we solve θ = cos 1 ( 3 13 2 ) on a calculator, since the domain of the inverse cosine function is [ 1 , 1 ] . However, there is a second solution:

cos 1 ( 3 + 13 2 )    1.26

This terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is

2 π cos 1 ( 3 + 13 2 )    5.02
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask