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Solving a linear equation involving the sine function

Find all possible exact solutions for the equation sin t = 1 2 .

Solving for all possible values of t means that solutions include angles beyond the period of 2 π . From [link] , we can see that the solutions are t = π 6 and t = 5 π 6 . But the problem is asking for all possible values that solve the equation. Therefore, the answer is

t = π 6 ± 2 π k   and   t = 5 π 6 ± 2 π k

where k is an integer.

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Given a trigonometric equation, solve using algebra .

  1. Look for a pattern that suggests an algebraic property, such as the difference of squares or a factoring opportunity.
  2. Substitute the trigonometric expression with a single variable, such as x or u .
  3. Solve the equation the same way an algebraic equation would be solved.
  4. Substitute the trigonometric expression back in for the variable in the resulting expressions.
  5. Solve for the angle.

Solve the linear trigonometric equation

Solve the equation exactly: 2 cos θ 3 = 5 , 0 θ < 2 π .

Use algebraic techniques to solve the equation.

2 cos θ 3 = 5 2 cos θ = 2 cos θ = 1 θ = π
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Solve exactly the following linear equation on the interval [ 0 , 2 π ) : 2 sin x + 1 = 0.

x = 7 π 6 , 11 π 6

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Solving equations involving a single trigonometric function

When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see [link] ). We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is π , not 2 π . Further, the domain of tangent is all real numbers with the exception of odd integer multiples of π 2 , unless, of course, a problem places its own restrictions on the domain.

Solving a problem involving a single trigonometric function

Solve the problem exactly: 2 sin 2 θ 1 = 0 , 0 θ < 2 π .

As this problem is not easily factored, we will solve using the square root property. First, we use algebra to isolate sin θ . Then we will find the angles.

2 sin 2 θ 1 = 0   2 sin 2 θ = 1 sin 2 θ = 1 2 sin 2 θ = ± 1 2 sin θ = ± 1 2 = ± 2 2 θ = π 4 , 3 π 4 , 5 π 4 , 7 π 4
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Solving a trigonometric equation involving cosecant

Solve the following equation exactly: csc θ = 2 , 0 θ < 4 π .

We want all values of θ for which csc θ = 2 over the interval 0 θ < 4 π .

csc θ = 2 1 sin θ = 2 sin θ = 1 2 θ = 7 π 6 , 11 π 6 , 19 π 6 , 23 π 6
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Solving an equation involving tangent

Solve the equation exactly: tan ( θ π 2 ) = 1 , 0 θ < 2 π .

Recall that the tangent function has a period of π . On the interval [ 0 , π ) , and at the angle of π 4 , the tangent has a value of 1. However, the angle we want is ( θ π 2 ) . Thus, if tan ( π 4 ) = 1 , then

θ π 2 = π 4 θ = 3 π 4 ± k π

Over the interval [ 0 , 2 π ) , we have two solutions:

θ = 3 π 4   and  θ = 3 π 4 + π = 7 π 4
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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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Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
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Oliver
proof for set theory
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Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
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Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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