Unit circle trigonometry | Lesson (article) | Khan Academy (2024)

The problems in this lesson involve circles and angle measures in radians, a unit for angle measure much like degrees. We can use radian measures to calculate arc lengths and sector areas, and we can calculate the sine, cosine, and tangent of radian measures.

In this lesson, we'll learn to:

1. Convert between radians and degrees
2. Use our knowledge of special right triangles to find radian measures
3. Identify the sine, cosine, and tangent of common radian measures

This lesson builds upon the following skills:

• Right triangle trigonometry
• Circle theorems

You can learn anything. Let's do this!

To convert between radians and degrees, we must be aware of the following information:

• The number of degrees of arc in a circle is $360$‍.
• The number of radians of arc in a circle is $2\pi$‍.

This means $360$‍ degrees is equivalent to $2\pi$‍ radians, and $180$‍ degrees is equivalent to $\pi$‍ radians. We can set up a proportional relationship to convert between radian and degree measures.

Example: Convert ${90}^{\circ }$‍ to radians.

This also means we can use radian measures to calculate arc lengths and sector areas just like we can with degree measures:

Example: In a circle with center $O$‍, central angle $AOB$‍ has a measure of ${\displaystyle \frac{2\pi }{3}}$‍ radians. The area of the sector formed by central angle $AOB$‍ is what fraction of the area of the circle?

At the beginning of each SAT math section, the following information about special right triangles is provided as reference:

These angle measures and their radian equivalents appear frequently in questions about circles and circle trigonometry. The table below shows the angles in special right triangles and their equivalent radian measures.

The radian measures we'll see on the SAT are usually multiples of the ones shown above.

On the test, we may be asked to find the radian measure of a central angle in a circle in the $xy$‍-plane, such as that of angle $AOB$‍ in the figure below. To do so, we'll draw a right triangle and look for the side length relationships in the special right triangles above.

We can draw a right triangle using the radius $\overline{OA}$‍ as the hypotenuse. Since one vertex of the right triangle is the origin, the two legs of the right triangle have lengths equal to the $x$‍- and $y$‍- coordinates of point $A$‍.

Since the two legs of the right triangle have the same length, we can conclude that it is a $45$‍-$45$‍-$90$‍ special right triangle, and the measure of angle $AOB$‍ must be ${45}^{\circ }$‍ or ${\displaystyle \frac{\pi }{4}}$‍ radians.

Trigonometry using radian measures is based on the unit circle, a circle centered on the origin with a radius of $1$‍.

We can describe each point $(x,y)$‍ on the circle and the slope of any radius in terms of $\theta$‍:

• $x=r\cos \theta =\cos \theta$‍
• $y=r\sin \theta =\sin \theta$‍
• ${\displaystyle \frac{y}{x}}=\tan \theta$‍

The table below shows the sine, cosine, and tangent of some common radian measures in the unit circle:

Note: If you already know these, that's great! If not, consider spending time on the more frequently-tested skills on the SAT before familiarizing yourself with the values of trigonometric functions.

We can describe each point $(x,y)$‍ on the unit circle and the slope of any radius in terms of $\theta$‍:

• $x=\cos \theta$‍
• $y=\sin \theta$‍
• ${\displaystyle \frac{y}{x}}=\tan \theta$‍