SAT — mathHard3 min read

Trigonometric Functions and the Unit Circle

Understand the unit circle and trig functions for the Digital SAT. Learn radian measure, key angle values, and co-function identities.

Tutor AI Team
Tutor AI Team

The unit circle extends trigonometry beyond right triangles to all angles. On the Digital SAT, you may encounter questions about radian measure, trig values at key angles, and co-function identities. This topic bridges basic SOH-CAH-TOA with more advanced trig concepts.

Core Concepts

The Unit Circle

A circle with radius 1 centred at the origin. A point at angle θ\theta has coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).

Radians

180°=π radians180° = \pi \text{ radians}

Conversion: degrees → radians: multiply by π180\frac{\pi}{180}. Radians → degrees: multiply by 180π\frac{180}{\pi}.

Key values:

Degrees Radians
0° 0
30° π6\frac{\pi}{6}
45° π4\frac{\pi}{4}
60° π3\frac{\pi}{3}
90° π2\frac{\pi}{2}
180° π\pi
360° 2π2\pi

Key Trig Values

Angle sin⁡\sin cos⁡\cos tan⁡\tan
0° 0 1 0
30° 12\frac{1}{2} 32\frac{\sqrt{3}}{2} 13\frac{1}{\sqrt{3}}
45° 22\frac{\sqrt{2}}{2} 22\frac{\sqrt{2}}{2} 1
60° 32\frac{\sqrt{3}}{2} 12\frac{1}{2} 3\sqrt{3}
90° 1 0 undefined

Co-function Identities

sin⁡θ=cos⁡(90°−θ)\sin\theta = \cos(90° - \theta) cos⁡θ=sin⁡(90°−θ)\cos\theta = \sin(90° - \theta)

Signs by Quadrant (ASTC)

  • Q1: All positive
  • Q2: Sin positive
  • Q3: Tan positive
  • Q4: Cos positive

Mnemonic: All Students Take Calculus.

Strategy Tips

Tip 1: Know the Key Values

Memorising sin/cos for 0°, 30°, 45°, 60°, 90° covers most SAT trig questions.

Tip 2: Co-function for Complementary Angles

sin⁡40°=cos⁡50°\sin 40° = \cos 50° because 40°+50°=90°40° + 50° = 90°.

Tip 3: Use Quadrant Signs

For angles beyond 90°, find the reference angle and apply the quadrant sign.

Worked Example: Example 1

Problem

Convert 150° to radians.

150×π180=5π6150 \times \frac{\pi}{180} = \frac{5\pi}{6}

Solution

Worked Example: Example 2

Problem

Find sin⁡(π3)\sin(\frac{\pi}{3}).

π3=60°\frac{\pi}{3} = 60°. sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}.

Solution

Worked Example: SAT-Style

Problem

If sin⁡x=0.6\sin x = 0.6, what is cos⁡(90°−x)\cos(90° - x)?

By co-function identity: cos⁡(90°−x)=sin⁡x=0.6\cos(90° - x) = \sin x = 0.6.

Solution

Worked Example: Example 4

Problem

In which quadrant is sin⁡θ<0\sin\theta < 0 and cos⁡θ>0\cos\theta > 0?

Quadrant IV (cos positive, sin negative).

Solution

Practice Problems

  1. Problem 1

    Convert 2π3\frac{2\pi}{3} to degrees.

    Problem 2

    Find cos⁡45°\cos 45°.

    Problem 3

    If cos⁡θ=0.8\cos\theta = 0.8, find sin⁡(90°−θ)\sin(90° - \theta).

    Problem 4

    In which quadrant is θ\theta if sin⁡θ>0\sin\theta > 0 and tan⁡θ<0\tan\theta < 0?

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Common Mistakes

  • Confusing degrees and radians. Check the mode on your calculator.
  • Not memorising the key values. These are tested frequently.
  • Getting quadrant signs wrong. Use ASTC to check signs.

Key Takeaways

  • ✓

    Unit circle: (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) for a point at angle θ\theta.

  • ✓

    Radians: π\pi radians = 180°.

  • ✓

    Co-function: sin⁡θ=cos⁡(90°−θ)\sin\theta = \cos(90° - \theta).

  • ✓

    ASTC gives the signs by quadrant.

  • ✓

    Memorise trig values for 0°, 30°, 45°, 60°, 90°.

Tags:
#sat#math#geometry#trigonometry#unit-circle

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