ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

The R Formula. How to write an expression in terms of sine or cosine only.

Updated on August 31, 2011

If you have an equation written in terms of sine and cosine, then the R formula can be used to rewrite the equation in terms of sine or cosine only. The 4 forms that you are rewriting the equations in are:

asinѲ + bcosѲ = Rsin(Ѳ + α)

asinѲ - bcosѲ = Rsin(Ѳ - α)

acosѲ +bsinѲ = Rcos (Ѳ - α)

acosѲ -bsinѲ = Rcos (Ѳ + α)

where R is a positive constant and α is a value between 0⁰ and 90⁰. Let’s take a look at a couple examples of using the R formula.

Example 1

Express √3sinѲ – cosѲ in the form Rsin(Ѳ-α), R>0 and 0≤α<90⁰, and solve the equation √3sinѲ – cosѲ = 0.5 in the range 0≤Ѳ<360⁰.

First of all use the addition formula,

sin(A-B) = sinAcosB – cosAsinB,

to expand Rsin(Ѳ-α):

Rsin(Ѳ-α) = R[sinѲcosα – cosѲsinα]

= RsinѲcosα – RcosѲsinα

So we now make this equal to √3sinѲ – cosѲ:

√3sinѲ – cosѲ = RsinѲcosα – RcosѲsinα

Now comparing coefficients on both sides of the equation you can write down 2 new equations:

Rcosα = √3 (Equation A)

Rsinα = 1 (Equation B)

Now if you square both of these equations and add both equations together you can find the value of R:

R²cos²α + R²sin²α = (√3)² + 1²

R²(cos²α + sin²α) = 4

Now you can get rid of the cos²α + sin²α as this is equal to 1 (using the identity cos²A + sin²A = 1)

R² = 4

R = 2

A quick way to work out R, is to square the two numbers from equations A and B, add them together, and take the square root (√[(√3)² + 1²] = √4 = 2 (a method similar to Pythagoras)

Now sub R= 2 back into Rcosα or Rsinα to find α:

2sinα = 1 (divide both sides by 2)

sinα = 0.5 (sin inverse both sides)

α = 30⁰

So √3sinѲ – cosѲ = 2sin(Ѳ-30)

Now we have written this in the R form we can now solve the equation:

2sin(Ѳ-30) = 0.5 (divide both sides by 2)

sin(Ѳ-30) = 0.25 (sin inverse both sides)

Ѳ - 30 = 14.5⁰, 165.5⁰ (make sure you find the other solution in the required range by subtracting the 14.5⁰ from 180⁰, and then add 30⁰ to both solutions)

Ѳ = 44.5⁰ or 195.5⁰.

Lets take a look at one more example of using the R formula:

Example 2

Write 5cosѲ + 3sinѲ in the form R(cosѲ-α).

First of all use the difference formula for cosine to expand R(cosѲ-α) ,

cos(A-B) = cosAcosB + sinAsinB so

Rcos(Ѳ-α) = R[cosѲcosα + sinѲsinα]

= RcosѲcosα + RsinѲsinα

So we now make this equal to 5cosѲ + 3sinѲ:

5cosѲ + 3sinѲ = RcosѲcosα + RsinѲsinα

Now comparing coefficients on both sides of the equation you can write down 2 new equations:

Rcosα = 5 (Equation A)

Rsinα = 3 (Equation B)

Now if you square both of these equations and add both equations together you can find the value of R or you can use the quick Pythagoras method as shown above.

√(5² + 3²) = √34

Now sub R= √34 back into Rcosα = 5 or Rsinα = 3 to find α:

√34sinα = 3 (divide both sides by 3)

sinα = 3/√34 (sin inverse both sides)

α = 31.0⁰

Therefore, 5cosѲ + 3sinѲ = √34(cosѲ-31.0) which you can now go on to use to solve equations.


Submit a Comment

No comments yet.


This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

Show Details
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)