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Course: Digital SAT Math > Unit 5

Lesson 5: Unit circle trigonometry: foundations
Test prep
Digital SAT Math
Foundations: Geometry and trigonometry
Unit circle trigonometry: foundations
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Unit circle trigonometry | Lesson

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A guide to unit circle trigonometry on the digital SAT

What are unit circle trigonometry problems?

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!

How do I convert between radians and degrees?

Radians & degrees

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Radians & degreesSee video transcript

Converting between radians and degrees

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 2π.
This means 360 degrees is equivalent to 2π radians, and 180 degrees is equivalent to π radians. We can set up a proportional relationship to convert between radian and degree measures.
radian measureπ=degree measure180∘
Example: Convert 90∘ to radians.
This also means we can use radian measures to calculate arc lengths and sector areas just like we can with degree measures:
central angle2π=arc lengthcircumference=sector areacircle area
Example: In a circle with center O, central angle AOB has a measure of 2π3 radians. The area of the sector formed by central angle AOB is what fraction of the area of the circle?

Try it!

try: compare radian and degree measures
Order the following angle measures from smallest to largest.
1

How do I use special right triangles to find radian measures?

Trig values of special angles

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Trig values of π/4See video transcript

Special right triangles in circles

At the beginning of each SAT math section, the following information about special right triangles is provided as reference:
A right triangle has angles measuring 30 degrees, 60 degrees, and 90 degrees. The length of the shorter leg is x, the length of the longer leg is x times the square root of 3, and the length of the hypotenuse is 2x.
A right triangle has angles measuring 90 degrees, 45 degrees, and 45 degrees. The lengths of the legs are both s, and the length of the hypotenuse is s times the square root of 2.
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.
Degree measureRadian measure
30∘π6
45∘π4
60∘π3
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.
A circle in the xy-plane has a center at the origin. OA is a radius of the circle, and the coordinates of point A are (1, 3^0.5). OB is also a radius of the circle lies on the positive x-axis.
We can draw a right triangle using the radius 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.
A right triangle with hypotenuse OA, a leg along the positive x-axis, and a vertical leg is drawn on top of the circle. The legs of the right triangle have lengths of 2.
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∘ or π4 radians.

Try it!

try: recognize a special right triangle in a circle
A circle in the xy-plane has a center at the origin. OA is a radius of the circle, and point A is in quadrant I. OB is also a radius of the circle lies on the positive x-axis.
In the figure above, O is the center of a circle in the xy-plane. The measure of angle AOB is π6 radians.
If the x-coordinate of point A is 23, what is its y-coordinate?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the radius of the circle?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

How do I find the sine, cosine, and tangent of radian measures?

Unit circle definition of trig functions

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Unit circleSee video transcript

The trig functions & right triangle trig ratios

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The trig functions & right triangle trig ratiosSee video transcript

Trigonometry using radian measures

Trigonometry using radian measures is based on the unit circle, a circle centered on the origin with a radius of 1.
A circle with a center at the origin and a radius of 1 is shown in the xy-plane. A radius connects the center of the circle and point (x, y) on the circle in the first quadrant. This radius forms an angle with the positive x-axis with measure theta.
We can describe each point (x,y) on the circle and the slope of any radius in terms of θ:
  • x=rcos⁡θ=cos⁡θ
  • y=rsin⁡θ=sin⁡θ
  • yx=tan⁡θ
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.
θx or cos⁡θy or sin⁡θtan⁡θ
0100
π6321233
π422221
π312323
π201undefined
2π3−1232−3
3π4−2222−1
π−100

Your turn!

practice: convert degrees to radians
The number of radians in a 135-degree angle can be written as aπ, where a is a constant. What is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1/2 or 6/10
  • an improper fraction, like 10/7 or 14/8
  • an exact decimal, like 0.75

practice: use special right triangle to find radian measure
A circle in the xy-plane has a center at the origin. OA is a radius of the circle, and the coordinates of point A are (1, 3^0.5). OB is also a radius of the circle lies on the positive x-axis.
In the xy-plane above, O is the center of the circle, and the measure of ∠AOB is πa. What is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Things to remember

radian measureπ=degree measure180∘
We can describe each point (x,y) on the unit circle and the slope of any radius in terms of θ:
  • x=cos⁡θ
  • y=sin⁡θ
  • yx=tan⁡θ

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