The Unit Circle

In this episode of TuesdACT, we take a look at the famous unit circle: a cool little circle that is often featured on ACT trig questions, so it’s a must-know ACT math thing. Check out the video for everything you need to know! (A few of the highlights are detailed below).

THIS is a unit circle. It’s a circle with radius of 1 centered about the origin.

There are a lot of fascinating aspects to the unit circle: I suggest you consult the interwebs or your math teacher to find out more. We’re just going to go through the absolute basics here that will help you get some ACT trig questions right. And if you haven’t already checked our post and video on sine, cosine, tangent, and SOHCAHTOA, you should probably do that first, it will make this all much easier to understand.

The ACT will test whether you know where angles larger than 360 degrees lie, and the unit circle helps us visualize this.

There are 360 degrees in a circle, but we can just keep swinging the arm of the angle around counterclockwise (just like the hands of the clock) to get to an angle bigger than 360. So, for example, if you want to know where an angle of 760 would be, you would circle around the circle twice (for a total of 720 degrees) and we would have 40 leftover degrees. So that angle would lie in the upper right quadrant of the unit circle (Quadrant I).

The ACT will also often use radians on trig questions, and the unit circle helps us wrap our heads around this.

You should know that:

90 degrees on the circle = 𝛑/2
180 degrees = 𝛑
270 degrees = 3𝛑/2
360 degrees =2𝛑

The ACT will also test whether you know where the sine, cosine, and tangent of angles are positive or negative on the unit circle.

Check out the video for the actual math explanation of why sine, cosine, and tangent are positive or negative in certain quadrants!

There’s a great mnemonic to help you remember where trig functions are positive or negative:

All Students Take Calculus

This helps you remember that:

In Quadrant 1 → All (sine, tangent, cosine) are positive
in Quadrant 2 → only Sine is positive (and cos and tan are negative)
in Quadrant 3 → only Tangent is positive (and sin and cos are negative)
in Quadrant 4 → only Cosine is positive (and sin and tan are negative)

ACT Trig Test Question:

Now let’s look at a test example to show you how this helps you out on a frequently-occurring ACT question:

If the value of cos x = -0.385, which of the following could be true about x?

A. 0 ≤ x < 𝛑
B. 𝛑/6 ≤ x < 𝛑/3
C. 𝛑/3 ≤ x < 𝛑/2
D. 𝛑/2 ≤ x < 2𝛑/3
E. 2𝛑/3 ≤ x ≤ 2𝛑

Using ASTC (All Students Take Calculus), we can figure out where cosine is negative and narrow down our answer choices. Cosine is negative in Q2 and Q3 and positive in Q1 and Q4. So we can eliminate any values that would fall in either Q1 or Q4. Answer choices A, B and C all have values that fall in Q1. Answer choice E is Q4. So that means the answer HAS to be D because that is the only answer choice in Q2 where cosine is negative. (And yes, it often is as easy as that on the ACT!)