Home / Articles / Flash Game Design: Trigonometry 101

Flash Game Design: Trigonometry 101

Chapter Description

Brush up on the basics of trigonometry, which should handle almost every Flash game development need you'll have.

The Heart of Trig

Is it all coming back to you yet? I hope so, because here's where we get to the real inner workings of trigonometry—where you can see how it's all going to come together. In this section we will cover the sine, cosine, and tangent functions, as well as projection. With knowledge of these operations under your belt, you will be able to understand the programming concepts you'll encounter in the following chapters (especially Chapter 6, "Collision Reactions").

Sine, Cosine, and Tangent

Sine, cosine, and tangent are known as trigonometric functions. Although what they mean is very simple, many people have trouble understanding them. This conceptual problem happens because it is easy to think that the trigonometric functions give a result by some esoteric or even mystical process. The truth is that these functions just use various ratios of the triangle side lengths to give results. Look at the triangle in the figure below. Notice that we are using x and y instead of a and b to label the side lengths. These are more common side names in programming. Notice the angle in the figure labeled angle.

Figure 13

All three of the trigonometric functions are defined by taking ratios of the sides of this triangle. A trigonometric function takes an angle (in Flash it must be measured in radians) and returns a value. For instance, the sine of 45° is .707. To test this for yourself in Flash, here is an ActionScript code snippet you can write:


Line 1 in the above code block sets the angle in degrees. Line 2 converts degrees to radians, and line 3 computes the sine and displays the result in the output window.

Table 3.1 lists these "big three" functions, their definitions, which methods of the Math object they correspond to in Flash, and a valid value range that can be returned from these functions.

TABLE 3.1 Trigonometric Functions in Flash

Trigonometric Function

Mathematica Definitionl

Method in Flash (Angle is in radians)

Minimum Result

Maximum Result














Negative infinity

Positive infinity

It wouldn't hurt to commit some simple results of the trigonometric functions to memory. This can help tremendously when debugging a script. Table 3.2 shows some simple values for you to remember, should you choose to.

TABLE 3.2 Trigonometric Equivalents

Typical Angles in Degrees




















Since you are able to calculate the sine, cosine, and tangent of an angle, it makes sense that there would also be some way to go from a number back to an angle. There is a set of functions for this, called the inverse trigonometric functions: inverse sine, inverse cosine, and inverse tangent. Some people use the term arc (as in arcsine) rather than inverse. Table 3.3 contains a list of the available inverse trigonometric functions.

TABLE 3.3 Inverse Trigonometric Functions

Inverse Trigonometric Function

Method in Flash


Inverse sine


Returns the angle whose sine is equal to the number

Inverse cosine


Returns the angle whose cosine is equal to the number

Inverse tangent


Returns the angle whose tangent is equal to the number

Inverse tangent2

Math.atan2(y, x)

Returns the angle whose tangent is equal to y/x

The inverse trigonometric functions take a number as an input parameter and return an angle in radians. To convince yourself of how this works, try this example in Flash:


Line 1 sets a variable called input with a value of .707. Line 2 uses the inverse sine method of the Math object (which returns an angle in radians) and then converts it to degrees. The result is traced in the Output window and should be very close to 45°. (It is not exactly 45° because the true sine of 45° has many more decimal places than .707.)

Figure 14


The word projection in the context of trigonometry means to project a quantity (such as distance or velocity) onto the x-axis and y-axis. Using what you'll learn in this section will help you when building games. For an example of what projection can help you accomplish, open the file shooter.fla in the Chapter03 folder on the CD-ROM. In this file, a ship rotates to point toward your mouse. When you click anywhere on the movie's stage, a projectile fires from the nose of the ship. The velocity of the projectile points toward your mouse (or at least to the place where your mouse was when you clicked it). In order for this movement to be programmed in Flash, the velocity must be projected along the x-axis and y-axis.

Figure 15

The programmatic movement seen in this example file is not covered until Chapter 4, "Basic Physics."

Imagine a diagonal line of length len drawn in Flash at angle ang. A piece of this line extends along the x-axis and another piece of it along the y-axis. If the angle were 0°, then the line would extend only along the x-axis. If the angle were 90° then the line would extend only along the y-axis. With any other angle, the line extends both in the x direction and the y direction. (Put another way, no two coordinates on the line have the same x or y value: A horizontal line always has the same y value for all of its coordinates. A vertical line always has the same x value for all of its coordinates. A diagonal line never repeats an x or y coordinate.) If you were to draw a right triangle from this diagonal line, then the two other sides of that triangle would be the pieces that extend along the x-axis and y-axis.

Finding the length of either (or both) of those pieces by using the values ang and len is called projection. These values are found by using the trigonometric functions that we've already discussed above.

Figure 16

As seen in the previous section:


In the example here, angle is replaced with ang and c with len. So:


To find the projection of len along the x-axis, we solve for x:


Or with ActionScript:


To find the y projection we use


And solve for y:


Which converts to this in ActionScript:


Think of projection like a shadow cast from an object onto the floor or a wall. For the example given in this section, first we would imagine a light source coming from below to cast a shadow on the x-axis. The length of the shadow cast from the line on the x-axis is the same as the projection we would calculate using trigonometry. Next we would imagine a light source coming from far off to the right shining left. The shadow cast on the y-axis is equal to that which we would calculate using trigonometry.

Figure 17

Adobe Press Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from Adobe Press and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about Adobe Press products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites; develop new products and services; conduct educational research; and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email ask@peachpit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by Adobe Press. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.adobepress.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020