<< Chapter < Page Chapter >> Page >

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Got questions? Get instant answers now!

Algebraic

For the following exercises, rewrite the product as a sum or difference.

16 sin ( 16 x ) sin ( 11 x )

8 ( cos ( 5 x ) cos ( 27 x ) )

Got questions? Get instant answers now!

20 cos ( 36 t ) cos ( 6 t )

Got questions? Get instant answers now!

2 sin ( 5 x ) cos ( 3 x )

sin ( 2 x ) + sin ( 8 x )

Got questions? Get instant answers now!

10 cos ( 5 x ) sin ( 10 x )

Got questions? Get instant answers now!

sin ( x ) sin ( 5 x )

1 2 ( cos ( 6 x ) cos ( 4 x ) )

Got questions? Get instant answers now!

For the following exercises, rewrite the sum or difference as a product.

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 ( 2 ) sin ( 3 )

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 ( 9 )

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 expressions is a true 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 and 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

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask