ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Trigonometric Addition and Difference Formulas (Identities) Also double angle formulas.

Updated on August 17, 2011

The addition and difference formulas used in trigonometry are as follows:

sin (A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB – cosAsinB

cos(A+B) = cosAcosB – sinAsinB

cos(A-B) = cosAcosB + sinAsinB

tan(A+B) = (tanA + tanB) ÷ (1 – tanAtanB)

tan(A-B) = (tanA - tanB) ÷ (1 + tanAtanB)

These formulas can be helpful in a variety of situations, one being in writing down exact values of sinA, cosA or tanA, and also deriving other trigonometric formulas such as the double angle formulas.

Example 1

Work out an exact solution to sin15⁰.

If you change 15 to 45 – 30, then the second trigonometric addition formula from the list above can be used to work out the exact solution to sin15⁰:

sin(A-B) = sinAcosB – cosAsinB

sin(45-30) = sin45cos30-cos45sin30

Now the values of sin45, cos30, cos45 and sin30 should already be known. If you don’t know them then these can be worked out by drawing out some right angle triangle. These are:

sin45 = (1/√2)

cos30 = (√3/2)

cos45 = (1/√2)

sin30 = ½


sin(45-30) = (1/√2)× (√3/2)- (1/√2) ½

= (√3/2√2) – (1/2√2)

= (√3 -1)/2√2

Let’s take a look at a slightly different example of using these addition/difference trigonometric identities:

Example 2

Write cos6xcosx + sin6xsinx as a single trigonometric function.

Here A = 6x and B =x, so all you need to do is substitute these values into the 4th formula listed at the top of this page:

cosAcosB + sinAsinB

= cos6xcosx + sin 6xsinx

= cos(6x-x)

= cos5x

You can also work out the double angle formula in a similar way, you can do this by changing the B’s to A’s in the 1st, 3rd and 5th formulas. Let me demonstrate this in the first trigonometric addition formula as the other double angle formulas will be covered in a later hubpage:

sin(A+B) = sinAcosB + cosAsinB

Now, change all of the B’s to A’s in the formula above:

sin(A +A) = sinAcosA + cosAsinA

Next, you can simplify this expression to give:

Sin(2A) = 2sinAcosA

This is known as the double angle formula for sin2A.


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)