SAT Math – Geometry and Trigonometry
Unit Circle Trigonometry
Understanding radian measure, reference angles, and trigonometric function values
The unit circle is a circle with radius 1 centered at the origin, providing a geometric foundation for trigonometric functions beyond right triangles. On the SAT, you'll convert between degrees and radians, determine trigonometric values at key angles, use reference angles to find values in all quadrants, and understand angle signs based on quadrant positions.
Success requires memorizing special angle values, understanding radian measure, recognizing reference angle patterns, and knowing which functions are positive in each quadrant. Unit circle concepts aren't just theoretical—they model periodic phenomena like sound waves, seasonal cycles, circular motion, alternating current, and any repeating pattern involving rotation or oscillation.
Understanding the Unit Circle
Unit Circle Definition
A circle with radius 1 centered at origin (0, 0).
Point (x, y): \(x = \cos(\theta)\), \(y = \sin(\theta)\)
Key insight: Coordinates of point at angle θ give trig values
Distance: Every point is exactly 1 unit from origin
Radian Measure
Alternative to degrees, measuring angles by arc length on unit circle.
Full circle: \(2\pi\) radians = 360°
Half circle: \(\pi\) radians = 180°
Conversion: Degrees × \(\frac{\pi}{180}\) = radians; Radians × \(\frac{180}{\pi}\) = degrees
Reference Angles
Acute angle between terminal side and x-axis.
Always: Between 0° and 90° (or 0 and \(\frac{\pi}{2}\))
Method: Measure shortest angle to x-axis
Sign: Determined by quadrant (magnitude from reference angle)
Quadrant Signs (All Students Take Calculus)
Mnemonic for remembering which functions are positive in each quadrant.
Quadrant II (90° to 180°): Sine positive
Quadrant III (180° to 270°): Tangent positive
Quadrant IV (270° to 360°): Cosine positive
Essential Values and Conversions
Common Angle Conversions
30° = \(\frac{\pi}{6}\) radians
45° = \(\frac{\pi}{4}\) radians
60° = \(\frac{\pi}{3}\) radians
90° = \(\frac{\pi}{2}\) radians
180° = \(\pi\) radians
Key Angle Values (Degrees)
0°: \(\sin(0°) = 0\), \(\cos(0°) = 1\), \(\tan(0°) = 0\)
30°: \(\sin(30°) = \frac{1}{2}\), \(\cos(30°) = \frac{\sqrt{3}}{2}\), \(\tan(30°) = \frac{1}{\sqrt{3}}\)
45°: \(\sin(45°) = \frac{\sqrt{2}}{2}\), \(\cos(45°) = \frac{\sqrt{2}}{2}\), \(\tan(45°) = 1\)
60°: \(\sin(60°) = \frac{\sqrt{3}}{2}\), \(\cos(60°) = \frac{1}{2}\), \(\tan(60°) = \sqrt{3}\)
90°: \(\sin(90°) = 1\), \(\cos(90°) = 0\), \(\tan(90°)\) undefined
Unit Circle Coordinates
At angle θ, point on unit circle is (\(\cos(\theta)\), \(\sin(\theta)\))
x-coordinate: Always cosine value
y-coordinate: Always sine value
Tangent: \(\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\)
Finding Reference Angles
Quadrant I: Reference angle = angle itself
Quadrant II: Reference angle = 180° - angle
Quadrant III: Reference angle = angle - 180°
Quadrant IV: Reference angle = 360° - angle
Common Pitfalls & Expert Tips
❌ Confusing sine and cosine values
sin(30°) = ½, but sin(60°) = √3/2. Don't mix them up! Memorize both or use reference triangles.
❌ Wrong signs in different quadrants
Remember "All Students Take Calculus" for quadrant signs. In Quadrant II, only sine is positive!
❌ Incorrect radian-degree conversion
To convert degrees to radians, multiply by π/180, not just π. 90° = 90 × π/180 = π/2, not 90π!
❌ Forgetting reference angles are always acute
Reference angle for 150° is 30° (not 150°). Always measure to closest x-axis, giving angle ≤ 90°.
✓ Expert Tip: Draw the unit circle
Sketch axes, mark quadrants, and plot the angle. Visual representation prevents sign errors!
✓ Expert Tip: Memorize special angles
Know 0°, 30°, 45°, 60°, 90° cold. These repeat in all quadrants with different signs!
✓ Expert Tip: Use complementary relationships
sin(30°) = cos(60°) and vice versa. Complementary angles (sum to 90°) swap sine and cosine!
Fully Worked SAT-Style Examples
Convert 120° to radians.
Solution:
Use conversion formula:
Radians = Degrees × \(\frac{\pi}{180}\)
\(120° \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3}\)
Answer: \(\frac{2\pi}{3}\) radians
Convert \(\frac{5\pi}{6}\) radians to degrees.
Solution:
Use conversion formula:
Degrees = Radians × \(\frac{180}{\pi}\)
\(\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = \frac{900}{6} = 150°\)
Answer: 150°
What is \(\sin(150°)\)?
Solution:
Identify quadrant:
150° is in Quadrant II (90° to 180°)
In Quadrant II, sine is positive
Find reference angle:
Reference angle = 180° - 150° = 30°
Apply reference angle:
\(\sin(150°) = +\sin(30°) = \frac{1}{2}\)
Answer: \(\frac{1}{2}\)
What is \(\cos(240°)\)?
Solution:
Identify quadrant:
240° is in Quadrant III (180° to 270°)
In Quadrant III, cosine is negative
Find reference angle:
Reference angle = 240° - 180° = 60°
Apply reference angle with sign:
\(\cos(240°) = -\cos(60°) = -\frac{1}{2}\)
Answer: \(-\frac{1}{2}\)
What is \(\tan(315°)\)?
Solution:
Identify quadrant:
315° is in Quadrant IV (270° to 360°)
In Quadrant IV, tangent is negative
Find reference angle:
Reference angle = 360° - 315° = 45°
Apply reference angle with sign:
\(\tan(315°) = -\tan(45°) = -1\)
Answer: -1
What is \(\sin\left(\frac{7\pi}{6}\right)\)?
Solution:
Convert to degrees (optional):
\(\frac{7\pi}{6} \times \frac{180}{\pi} = \frac{7 \times 180}{6} = 210°\)
Identify quadrant:
210° is in Quadrant III
In Quadrant III, sine is negative
Find reference angle:
Reference = 210° - 180° = 30° (or \(\frac{\pi}{6}\))
\(\sin\left(\frac{7\pi}{6}\right) = -\sin(30°) = -\frac{1}{2}\)
Answer: \(-\frac{1}{2}\)
If \(\cos(\theta) = \frac{3}{5}\) and θ is in Quadrant I, what is \(\sin(\theta)\)?
Solution:
Use Pythagorean identity:
\(\sin^2(\theta) + \cos^2(\theta) = 1\)
\(\sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1\)
\(\sin^2(\theta) + \frac{9}{25} = 1\)
\(\sin^2(\theta) = \frac{16}{25}\)
Determine sign:
In Quadrant I, sine is positive
\(\sin(\theta) = +\frac{4}{5}\)
Answer: \(\frac{4}{5}\)
Quick Reference Chart
Quadrant Signs
I: All positive
II: Sin positive
III: Tan positive
IV: Cos positive
Key Conversions
π rad = 180°
π/2 rad = 90°
π/3 rad = 60°
π/4 rad = 45°
Unit Circle Trigonometry: Beyond Right Triangles
The unit circle extends trigonometry beyond right triangles, defining sine, cosine, and tangent for all angles through coordinates on a circle of radius one—a generalization essential for modeling periodic phenomena, circular motion, and wave behavior. The SAT tests unit circle concepts because they represent mathematical sophistication necessary for calculus, physics, engineering, and any field involving rotation, oscillation, or cyclical patterns. Understanding that any point at angle θ on the unit circle has coordinates (cos(θ), sin(θ)) connects trigonometric functions to geometric positions, while recognizing tangent as the ratio sin/cos reveals why tangent is undefined when cosine equals zero. Radian measure—defining angles by arc length rather than arbitrary degree divisions—provides the natural unit for circular motion, with 2π radians representing one full rotation because circumference of unit circle equals 2π. Reference angles enable calculating trigonometric values in any quadrant by reducing to acute angles, while the quadrant-sign mnemonic "All Students Take Calculus" systematically determines function signs based on position. The Pythagorean identity sin²(θ) + cos²(θ) = 1 emerges naturally from the distance formula on the unit circle, providing powerful relationships among trigonometric functions. These concepts transcend classroom exercises, empowering you to analyze alternating current in electrical engineering (voltage and current varying sinusoidally), understand simple harmonic motion in physics (pendulums and springs oscillating with trigonometric displacement), model seasonal temperature variations, calculate planetary positions in orbital mechanics, and recognize that musical notes involve trigonometric waveforms. Every conversion between radians and degrees, every reference angle calculation, every determination of sign based on quadrant represents facility with the mathematical language describing rotation, periodicity, and circular relationships—skills fundamental to advanced mathematics and scientific modeling of rhythmic, cyclical, or rotational phenomena from heartbeats to electromagnetic waves.