<< Chapter < Page Chapter >> Page >

cos ( 6 t ) + cos ( 4 t )

2 cos ( 5 t ) cos t

Got questions? Get instant answers now!

sin ( 3 x ) + sin ( 7 x )

Got questions? Get instant answers now!

cos ( 7 x ) + cos ( 7 x )

2 cos ( 7 x )

Got questions? Get instant answers now!

sin ( 3 x ) sin ( 3 x )

Got questions? Get instant answers now!

cos ( 3 x ) + cos ( 9 x )

2 cos ( 6 x ) cos ( 3 x )

Got questions? Get instant answers now!

sin h sin ( 3 h )

Got questions? Get instant answers now!

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

cos ( 45° ) cos ( 15° )

1 4 ( 1 + 3 )

Got questions? Get instant answers now!

cos ( 45° ) sin ( 15° )

Got questions? Get instant answers now!

sin ( −345° ) sin ( −15° )

1 4 ( 3 2 )

Got questions? Get instant answers now!

sin ( 195° ) cos ( 15° )

Got questions? Get instant answers now!

sin ( −45° ) sin ( −15° )

1 4 ( 3 1 )

Got questions? Get instant answers now!

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

cos ( 23° ) sin ( 17° )

Got questions? Get instant answers now!

2 sin ( 100° ) sin ( 20° )

cos ( 80° ) cos ( 120° )

Got questions? Get instant answers now!

2 sin ( −100° ) sin ( −20° )

Got questions? Get instant answers now!

sin ( 213° ) cos ( )

1 2 ( sin ( 221° ) + sin ( 205° ) )

Got questions? Get instant answers now!

2 cos ( 56° ) cos ( 47° )

Got questions? Get instant answers now!

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

sin ( 76° ) + sin ( 14° )

2 cos ( 31° )

Got questions? Get instant answers now!

cos ( 58° ) cos ( 12° )

Got questions? Get instant answers now!

sin ( 101° ) sin ( 32° )

2 cos ( 66.5° ) sin ( 34.5° )

Got questions? Get instant answers now!

cos ( 100° ) + cos ( 200° )

Got questions? Get instant answers now!

sin ( −1° ) + sin ( −2° )

2 sin ( −1.5° ) cos ( 0.5° )

Got questions? Get instant answers now!

For the following exercises, prove the identity.

cos ( a + b ) cos ( a b ) = 1 tan a tan b 1 + tan a tan b

Got questions? Get instant answers now!

4 sin ( 3 x ) cos ( 4 x ) = 2 sin ( 7 x ) 2 sin x

2 sin ( 7 x ) 2 sin x = 2 sin ( 4 x + 3 x ) 2 sin ( 4 x 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) 2 ( sin ( 4 x ) cos ( 3 x ) sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x )

Got questions? Get instant answers now!

6 cos ( 8 x ) sin ( 2 x ) sin ( 6 x ) = −3 sin ( 10 x ) csc ( 6 x ) + 3

Got questions? Get instant answers now!

sin x + sin ( 3 x ) = 4 sin x cos 2 x

sin x + sin ( 3 x ) = 2 sin ( 4 x 2 ) cos ( 2 x 2 ) = 2 sin ( 2 x ) cos x = 2 ( 2 sin x cos x ) cos x = 4 sin x cos 2 x

Got questions? Get instant answers now!

2 ( cos 3 x cos x sin 2 x ) = cos ( 3 x ) + cos x

Got questions? Get instant answers now!

2 tan x cos ( 3 x ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) sin ( 2 x ) ) ) cos x = 1 cos x ( sin ( 4 x ) sin ( 2 x ) ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

Got questions? Get instant answers now!

cos ( a + b ) + cos ( a b ) = 2 cos a cos b

Got questions? Get instant answers now!

Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

cos ( 58° ) + cos ( 12° )

2 cos ( 35° ) cos ( 23° ) , 1.5081

Got questions? Get instant answers now!

sin ( ) sin ( )

Got questions? Get instant answers now!

cos ( 44° ) cos ( 22° )

2 sin ( 33° ) sin ( 11° ) , 0.2078

Got questions? Get instant answers now!

cos ( 176° ) sin ( )

Got questions? Get instant answers now!

sin ( −14° ) sin ( 85° )

1 2 ( cos ( 99° ) cos ( 71° ) ) , −0.2410

Got questions? Get instant answers now!

Technology

For the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

2 sin ( 2 x ) sin ( 3 x ) = cos x cos ( 5 x )

Got questions? Get instant answers now!

cos ( 10 θ ) + cos ( 6 θ ) cos ( 6 θ ) cos ( 10 θ ) = cot ( 2 θ ) cot ( 8 θ )

It is an identity.

Got questions? Get instant answers now!

sin ( 3 x ) sin ( 5 x ) cos ( 3 x ) + cos ( 5 x ) = tan x

Got questions? Get instant answers now!

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

It is not an identity, but 2 cos 3 x is.

Got questions? Get instant answers now!

sin ( 2 x ) + sin ( 4 x ) sin ( 2 x ) sin ( 4 x ) = tan ( 3 x ) cot x

Got questions? Get instant answers now!

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

sin ( 9 t ) sin ( 3 t ) cos ( 9 t ) + cos ( 3 t )

tan ( 3 t )

Got questions? Get instant answers now!

2 sin ( 8 x ) cos ( 6 x ) sin ( 2 x )

Got questions? Get instant answers now!

sin ( 3 x ) sin x sin x

2 cos ( 2 x )

Got questions? Get instant answers now!

cos ( 5 x ) + cos ( 3 x ) sin ( 5 x ) + sin ( 3 x )

Got questions? Get instant answers now!

sin x cos ( 15 x ) cos x sin ( 15 x )

sin ( 14 x )

Got questions? Get instant answers now!

Extensions

For the following exercises, prove the following sum-to-product formulas.

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

Got questions? Get instant answers now!

cos x + cos y = 2 cos ( x + y 2 ) cos ( x y 2 )

Start with cos x + cos y . Make a substitution and let x = α + β and let y = α β , so cos x + cos y becomes cos ( α + β ) + cos ( α β ) = cos α cos β sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β

Since x = α + β and y = α β , we can solve for α and β in terms of x and y and substitute in for 2 cos α cos β and get 2 cos ( x + y 2 ) cos ( x y 2 ) .

Got questions? Get instant answers now!

For the following exercises, prove the identity.

sin ( 6 x ) + sin ( 4 x ) sin ( 6 x ) sin ( 4 x ) = tan ( 5 x ) cot x

Got questions? Get instant answers now!

cos ( 3 x ) + cos x cos ( 3 x ) cos x = cot ( 2 x ) cot x

cos ( 3 x ) + cos x cos ( 3 x ) cos x = 2 cos ( 2 x ) cos x 2 sin ( 2 x ) sin x = cot ( 2 x ) cot x

Got questions? Get instant answers now!

cos ( 6 y ) + cos ( 8 y ) sin ( 6 y ) sin ( 4 y ) = cot y cos ( 7 y ) sec ( 5 y )

Got questions? Get instant answers now!

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = tan y

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y

Got questions? Get instant answers now!

sin ( 10 x ) sin ( 2 x ) cos ( 10 x ) + cos ( 2 x ) = tan ( 4 x )

Got questions? Get instant answers now!

cos x cos ( 3 x ) = 4 sin 2 x cos x

cos x cos ( 3 x ) = 2 sin ( 2 x ) sin ( x ) = 2 ( 2 sin x cos x ) sin x = 4 sin 2 x cos x

Got questions? Get instant answers now!

( cos ( 2 x ) cos ( 4 x ) ) 2 + ( sin ( 4 x ) + sin ( 2 x ) ) 2 = 4 sin 2 ( 3 x )

Got questions? Get instant answers now!

tan ( π 4 t ) = 1 tan t 1 + tan t

tan ( π 4 t ) = tan ( π 4 ) tan t 1 + tan ( π 4 ) tan ( t ) = 1 tan t 1 + tan t

Got questions? Get instant answers now!

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
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
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
Noven Reply
what's your topic about?
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
Practice Key Terms 2

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