Unit Circle Basics on the SAT
Coordinates of Special Angles
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Unit Circle Basics
The unit circle connects angles to exact trig values. On the SAT, you only need a handful of special angles, but understanding where those values come from makes them much easier to remember.
This lesson shows how the unit circle is built from right triangles, how to read coordinates as $(\cos \theta, \sin \theta)$, and how to keep track of signs in different quadrants.
A Simple Definition Unlocks Unit Circle Basics
The unit circle has radius $1$, so a point on the circle at angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$. That means the $x$ coordinate is cosine and the $y$ coordinate is sine.
The reference angles $30^\circ$, $45^\circ$, and $60^\circ$ give the exact values. The signs depend on the quadrant, so it helps to remember which quadrants have positive sine and cosine values.
Work Through Unit Circle Basics Step by Step
What are $\cos 45^\circ$ and $\sin 45^\circ$?
Use the $45^\circ$ triangle to find the unit circle coordinates step by step.
Use the $45^\circ$ triangle to get the cosine value.
Use the $45^\circ$ triangle to get the sine value.
Write the coordinate as $(\cos \theta, \sin \theta)$.
Use Desmos to Check Unit Circle Basics
What are $\cos 45^\circ$ and $\sin 45^\circ$?
Desmos can evaluate trig values quickly if you need a decimal. Use it when the question asks for approximations.
cos(45 * pi/180)
Exact values are usually preferred on the SAT for special angles.
Desmos can confirm signs and approximate values, but memorized exact values are faster and more reliable for SAT special angles.
Desmos is faster for quick sign checks or decimal confirmations, but memorized exact values are faster and more reliable for SAT special angles.
Expert move: Use Desmos to evaluate trig ratios or inverse trig when the angle is not special; keep exact values when possible and check the quadrant/sign.
- Desmos features used: trig evaluation.
- Common mistake: swapping sine and cosine coordinates.
Practice Unit Circle Basics with SAT-Style Questions
Recall special angle coordinates.
What are the coordinates of on the unit circle?
What is ?
What are the coordinates of ?
What are the coordinates of ?
Key Takeaways to Remember for Unit Circle Basics
- Unit circle coordinates are $(\cos \theta, \sin \theta)$.
- Memorize the special angles and their coordinates.
- Desmos is helpful for decimal checks when needed.
