<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Express products as sums.
  • Express sums as products.
Photo of the UCLA marching band.
The UCLA marching band (credit: Eric Chan, Flickr).

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, [link] represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

Graph of a sound wave for the musical note A - it is a periodic function much like sin and cos - from 0 to .01

Expressing products as sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas , which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

Expressing products as sums for cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine . If we add the two equations, we get:

cos α cos β + sin α sin β = cos ( α β ) + cos α cos β sin α sin β = cos ( α + β ) ________________________________ 2 cos α cos β = cos ( α β ) + cos ( α + β )

Then, we divide by 2 to isolate the product of cosines:

cos α cos β = 1 2 [ cos ( α β ) + cos ( α + β ) ]

Given a product of cosines, express as a sum.

  1. Write the formula for the product of cosines.
  2. Substitute the given angles into the formula.
  3. Simplify.

Writing the product as a sum using the product-to-sum formula for cosine

Write the following product of cosines as a sum: 2 cos ( 7 x 2 ) cos 3 x 2 .

We begin by writing the formula for the product of cosines:

cos α cos β = 1 2 [ cos ( α β ) + cos ( α + β ) ]

We can then substitute the given angles into the formula and simplify.

2 cos ( 7 x 2 ) cos ( 3 x 2 ) = ( 2 ) ( 1 2 ) [ cos ( 7 x 2 3 x 2 ) + cos ( 7 x 2 + 3 x 2 ) ]                             = [ cos ( 4 x 2 ) + cos ( 10 x 2 ) ]                             = cos 2 x + cos 5 x
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the product-to-sum formula to write the product as a sum or difference: cos ( 2 θ ) cos ( 4 θ ) .

1 2 ( cos 6 θ + cos 2 θ )

Got questions? Get instant answers now!

Expressing the product of sine and cosine as a sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine . If we add the sum and difference identities, we get:

                     sin ( α + β ) = sin α cos β + cos α sin β +                  sin ( α β ) = sin α cos β cos α sin β _________________________________________ sin ( α + β ) + sin ( α β ) = 2 sin α cos β

Then, we divide by 2 to isolate the product of cosine and sine:

sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ]

Writing the product as a sum containing only sine or cosine

Express the following product as a sum containing only sine or cosine and no products: sin ( 4 θ ) cos ( 2 θ ) .

Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.

sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ] sin ( 4 θ ) cos ( 2 θ ) = 1 2 [ sin ( 4 θ + 2 θ ) + sin ( 4 θ 2 θ ) ] = 1 2 [ sin ( 6 θ ) + sin ( 2 θ ) ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the product-to-sum formula to write the product as a sum: sin ( x + y ) cos ( x y ) .

1 2 ( sin 2 x + sin 2 y )

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com 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