<< 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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
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?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

Get the best Algebra and trigonometry course in your pocket!





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