Copyright

Circular Functions: Equations & Examples

Circular Functions: Equations & Examples
Coming up next: Finding the Area of a Sector: Formula & Practice Problems

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 What Are Circular Functions?
  • 1:32 Using Trigonometry to…
  • 3:41 Trigonometry & the…
  • 4:48 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Betty Bundly

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

In this lesson, we will learn how to define the basic trigonometric functions using a circle. In doing so, every angle in a circle will be paired with an ordered (x,y) point in the coordinate plane using the basic trigonometric functions.

What Are Circular Functions?

You may have initially learned about the sine, cosine and tangent of an angle as the ratio of the sides of a right triangle. The sides that form the right angle are called legs, and the third side is called the hypotenuse. Given angle A, the three ratios are as follows:

  • sinA = (length of side opposite angle A) / length of hypotenuse)
  • cosA = (length of side adjacent to angle A) / (length of hypotenuse)
  • tanA = (length of side opposite angle A) / (length of side adjacent to angle A) = sinA / cosA

Right Triangle

The circular functions may be thought of as a way to extend these extremely useful mathematical relationships to any triangle. To visualize circular functions, we first start with a unit circle, or a circle with a radius equal to one unit of measurement.

Circle Radius 1

In this circle, draw an x-y coordinate plane, with the origin at the center of the circle.

Unit Circle

Now, insert a right triangle in Quadrant I, so that the hypotenuse equals the radius of the circle, one leg of the right angle lies horizontally along the x-axis and the other leg is a vertical line that meets the hypotenuse lying on the circle. The angle will be formed by some length of the x-axis on one side and the hypotenuse on the other. The point of the circle where the hypotenuse and the vertical leg intersect is a rectangular coordinate point (x,y).

Right Triangle in circle

Using Trigonometry to Describe Points on a Circle

How does this x and y value translate into trigonometry? Since the hypotenuse is always equal to the radius of the circle, we call it r. Since the horizontal leg lies along the x-axis, we call it x. Since the vertical leg describes a position on the y-axis, we call it y. Using x, y, and r, we can now describe sinA, cosA as follows:

  • SinA = y / r
  • CosA= x / r

Now, to solve for x and y, multiply both sides of sinA = y / r by r and multiply cosA= x / r by r.

y = r * sinA

If we multiply both side of cosA = y / r by r, we obtain:

x = r * cosA

Now, as the hypotenuse moves counterclockwise around the circumference of the circle, a right angle can be formed this same way for every angle from 0 degrees to 360 degrees. Each of these angles can be described by the point (x,y) or (rcosA, rsinA).

(rcosA, rsinA)

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support