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Find all solutions for tan x = 3 .

π 3 ± π k

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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 .

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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
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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 .

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Solve cos θ = 0.2.

θ 1.7722 ± 2 π k and θ 4.5110 ± 2 π k

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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
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Questions & Answers

A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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