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Algebraic

For the following exercises, find all solutions exactly on the interval 0 θ < 2 π .

2 sin θ = 2

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2 sin θ = 3

π 3 , 2 π 3

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2 cos θ = 2

3 π 4 , 5 π 4

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tan x = 1

π 4 , 5 π 4

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4 sin 2 x 2 = 0

π 4 , 3 π 4 , 5 π 4 , 7 π 4

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For the following exercises, solve exactly on [ 0 , 2 π ) .

2 cos θ = 2

π 4 , 7 π 4

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2 sin θ = −1

7 π 6 , 11 π 6

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2 sin θ = 3

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2 sin ( 3 θ ) = 1

π 18 , 5 π 18 , 13 π 18 , 17 π 18 , 25 π 18 , 29 π 18

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2 cos ( 3 θ ) = 2

3 π 12 , 5 π 12 , 11 π 12 , 13 π 12 , 19 π 12 , 21 π 12

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2 sin ( π θ ) = 1

1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6

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2 cos ( π 5 θ ) = 3

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For the following exercises, find all exact solutions on [ 0 , 2 π ) .

sec ( x ) sin ( x ) 2 sin ( x ) = 0

0 , π 3 , π , 5 π 3

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tan ( x ) 2 sin ( x ) tan ( x ) = 0

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2 cos 2 t + cos ( t ) = 1

π 3 , π , 5 π 3

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2 tan 2 ( t ) = 3 sec ( t )

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2 sin ( x ) cos ( x ) sin ( x ) + 2 cos ( x ) 1 = 0

π 3 , 3 π 2 , 5 π 3

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tan 2 ( x ) = −1 + 2 tan ( x )

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8 sin 2 ( x ) + 6 sin ( x ) + 1 = 0

π sin 1 ( 1 4 ) , 7 π 6 , 11 π 6 , 2 π + sin 1 ( 1 4 )

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For the following exercises, solve with the methods shown in this section exactly on the interval [ 0 , 2 π ) .

sin ( 3 x ) cos ( 6 x ) cos ( 3 x ) sin ( 6 x ) = −0.9

1 3 ( sin 1 ( 9 10 ) ) , π 3 1 3 ( sin 1 ( 9 10 ) ) , 2 π 3 + 1 3 ( sin 1 ( 9 10 ) ) , π 1 3 ( sin 1 ( 9 10 ) ) , 4 π 3 + 1 3 ( sin 1 ( 9 10 ) ) , 5 π 3 1 3 ( sin 1 ( 9 10 ) )

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sin ( 6 x ) cos ( 11 x ) cos ( 6 x ) sin ( 11 x ) = −0.1

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cos ( 2 x ) cos x + sin ( 2 x ) sin x = 1

0

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6 sin ( 2 t ) + 9 sin t = 0

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9 cos ( 2 θ ) = 9 cos 2 θ 4

π 6 , 5 π 6 , 7 π 6 , 11 π 6

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cos ( 2 t ) = sin t

3 π 2 , π 6 , 5 π 6

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cos ( 6 x ) cos ( 3 x ) = 0

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For the following exercises, solve exactly on the interval [ 0 , 2 π ) . Use the quadratic formula if the equations do not factor.

tan 2 x 3 tan x = 0

0 , π 3 , π , 4 π 3

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sin 2 x + sin x 2 = 0

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sin 2 x 2 sin x 4 = 0

There are no solutions.

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5 cos 2 x + 3 cos x 1 = 0

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3 cos 2 x 2 cos x 2 = 0

cos 1 ( 1 3 ( 1 7 ) ) , 2 π cos 1 ( 1 3 ( 1 7 ) )

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5 sin 2 x + 2 sin x 1 = 0

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tan 2 x + 5 tan x 1 = 0

tan 1 ( 1 2 ( 29 5 ) ) , π + tan 1 ( 1 2 ( 29 5 ) ) , π + tan 1 ( 1 2 ( 29 5 ) ) , 2 π + tan 1 ( 1 2 ( 29 5 ) )

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tan 2 x tan x 2 = 0

There are no solutions.

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For the following exercises, find exact solutions on the interval [ 0 , 2 π ) . Look for opportunities to use trigonometric identities.

sin 2 x cos 2 x sin x = 0

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sin 2 x + cos 2 x = 0

There are no solutions.

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sin ( 2 x ) sin x = 0

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cos ( 2 x ) cos x = 0

0 , 2 π 3 , 4 π 3

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2 tan x 2 sec 2 x sin 2 x = cos 2 x

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1 cos ( 2 x ) = 1 + cos ( 2 x )

π 4 , 3 π 4 , 5 π 4 , 7 π 4

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10 sin x cos x = 6 cos x

sin 1 ( 3 5 ) , π 2 , π sin 1 ( 3 5 ) , 3 π 2

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−3 sin t = 15 cos t sin t

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4 cos 2 x 4 = 15 cos x

cos 1 ( 1 4 ) , 2 π cos 1 ( 1 4 )

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8 sin 2 x + 6 sin x + 1 = 0

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8 cos 2 θ = 3 2 cos θ

π 3 , cos 1 ( 3 4 ) , 2 π cos 1 ( 3 4 ) , 5 π 3

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6 cos 2 x + 7 sin x 8 = 0

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12 sin 2 t + cos t 6 = 0

cos 1 ( 3 4 ) , cos 1 ( 2 3 ) , 2 π cos 1 ( 2 3 ) , 2 π cos 1 ( 3 4 )

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tan x = 3 sin x

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cos 3 t = cos t

0 , π 2 , π , 3 π 2

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Graphical

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 sin 2 x 5 sin x + 1 = 0

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8 cos 2 x 2 cos x 1 = 0

π 3 , cos −1 ( 1 4 ) , 2 π cos −1 ( 1 4 ) , 5 π 3

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100 tan 2 x + 20 tan x 3 = 0

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2 cos 2 x cos x + 15 = 0

There are no solutions.

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20 sin 2 x 27 sin x + 7 = 0

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2 tan 2 x + 7 tan x + 6 = 0

π + tan −1 ( −2 ) , π + tan −1 ( 3 2 ) , 2 π + tan −1 ( −2 ) , 2 π + tan −1 ( 3 2 )

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130 tan 2 x + 69 tan x 130 = 0

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Technology

For the following exercises, use a calculator to find all solutions to four decimal places.

sin x = 0.27

2 π k + 0.2734 , 2 π k + 2.8682

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tan x = −0.34

π k 0.3277

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For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval [ 0 , 2 π ) . Round to four decimal places.

tan 2 x + 3 tan x 3 = 0

0.6694 , 1.8287 , 3.8110 , 4.9703

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Questions & Answers

what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
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the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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1+cos²A/cos²A=2cosec²A-1
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test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
yeah
Morosi
prime number?
Morosi
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
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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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