<< Chapter < Page Chapter >> Page >

Algebraic

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

2 sin θ = 2

Got questions? Get instant answers now!

2 sin θ = 3

π 3 , 2 π 3

Got questions? Get instant answers now!

2 cos θ = 2

3 π 4 , 5 π 4

Got questions? Get instant answers now!

tan x = 1

π 4 , 5 π 4

Got questions? Get instant answers now!

4 sin 2 x 2 = 0

π 4 , 3 π 4 , 5 π 4 , 7 π 4

Got questions? Get instant answers now!

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

2 cos θ = 2

π 4 , 7 π 4

Got questions? Get instant answers now!

2 sin θ = −1

7 π 6 , 11 π 6

Got questions? Get instant answers now!

2 sin θ = 3

Got questions? Get instant answers now!

2 sin ( 3 θ ) = 1

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

Got questions? Get instant answers now!

2 cos ( 3 θ ) = 2

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

Got questions? Get instant answers now!

2 sin ( π θ ) = 1

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

Got questions? Get instant answers now!

2 cos ( π 5 θ ) = 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

2 cos 2 t + cos ( t ) = 1

π 3 , π , 5 π 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

π 3 , 3 π 2 , 5 π 3

Got questions? Get instant answers now!

tan 2 ( x ) = −1 + 2 tan ( x )

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 ) )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

9 cos ( 2 θ ) = 9 cos 2 θ 4

π 6 , 5 π 6 , 7 π 6 , 11 π 6

Got questions? Get instant answers now!

cos ( 2 t ) = sin t

3 π 2 , π 6 , 5 π 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

sin 2 x + sin x 2 = 0

Got questions? Get instant answers now!

sin 2 x 2 sin x 4 = 0

There are no solutions.

Got questions? Get instant answers now!

5 cos 2 x + 3 cos x 1 = 0

Got questions? Get instant answers now!

3 cos 2 x 2 cos x 2 = 0

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

Got questions? Get instant answers now!

5 sin 2 x + 2 sin x 1 = 0

Got questions? Get instant answers now!

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 ) )

Got questions? Get instant answers now!

tan 2 x tan x 2 = 0

There are no solutions.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

sin 2 x + cos 2 x = 0

There are no solutions.

Got questions? Get instant answers now!

sin ( 2 x ) sin x = 0

Got questions? Get instant answers now!

cos ( 2 x ) cos x = 0

0 , 2 π 3 , 4 π 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

π 4 , 3 π 4 , 5 π 4 , 7 π 4

Got questions? Get instant answers now!

10 sin x cos x = 6 cos x

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

Got questions? Get instant answers now!

−3 sin t = 15 cos t sin t

Got questions? Get instant answers now!

4 cos 2 x 4 = 15 cos x

cos 1 ( 1 4 ) , 2 π cos 1 ( 1 4 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

8 cos 2 θ = 3 2 cos θ

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

Got questions? Get instant answers now!

6 cos 2 x + 7 sin x 8 = 0

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

tan x = 3 sin x

Got questions? Get instant answers now!

cos 3 t = cos t

0 , π 2 , π , 3 π 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

8 cos 2 x 2 cos x 1 = 0

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

Got questions? Get instant answers now!

100 tan 2 x + 20 tan x 3 = 0

Got questions? Get instant answers now!

2 cos 2 x cos x + 15 = 0

There are no solutions.

Got questions? Get instant answers now!

20 sin 2 x 27 sin x + 7 = 0

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

130 tan 2 x + 69 tan x 130 = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

tan x = −0.34

π k 0.3277

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

Solve the problem 3x^2_2x+4=0
Ayoman Reply
can anyone recommend an app or. website to help me refresh my knowledge and abilities? I stopped learning more about calc and trig in the 90's
scott Reply
if 6x=-2 find value of 6/x
issaya Reply
l don't know
Aman
I don't know
Aman
l don't know
Aman
I don't know
Aman
x=.333333333334, so 6/x= 18±
scott
x=.33333333334± 6/x=18±
scott
that is strange, I don't remember inserting the À. the result I got was x= 3333333334± 6/x=18±
scott
I need glasses
scott
X=(-1÷3) X equals minus one third
Melvin
if sin15°=√p, express the following in terms of p
Norman Reply
I am learning and does someone have the time to tell me what or where this particular equation would be used?
scott
prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask