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AP Precalculus: Trigonometry

Master angles, ratios, unit circle, laws, and inverse functions

📐 Angles ⭕ Unit Circle 📊 Laws 🔄 Inverses

📚 Understanding Trigonometry

Trigonometry studies the relationships between angles and sides of triangles. These concepts extend to the unit circle, periodic functions, and solving real-world problems involving angles, distances, and waves. This guide covers all essential formulas and techniques for AP Precalculus.

1 Radian & Degree Conversion

Radians measure angles based on the radius of a circle. One full rotation = \(2\pi\) radians = \(360°\).

Degrees → Radians
\(\theta_{rad} = \theta° \times \frac{\pi}{180}\)
Radians → Degrees
\(\theta° = \theta_{rad} \times \frac{180}{\pi}\)

Arc Length & Sector Area

Arc Length
\(s = r\theta\)
(\(\theta\) in radians)
Sector Area
\(A = \frac{1}{2}r^2\theta\)
(\(\theta\) in radians)
📌 Example

Convert \(135°\) to radians:

\(135° \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}\) radians

2 Quadrants, Coterminal & Reference Angles

The coordinate plane is divided into four quadrants. Each quadrant has specific sign rules for trig functions.

Quadrant Ranges & Signs

Quadrant II
\(90° - 180°\)
sin + | cos − | tan −
Quadrant I
\(0° - 90°\)
sin + | cos + | tan +
Quadrant III
\(180° - 270°\)
sin − | cos − | tan +
Quadrant IV
\(270° - 360°\)
sin − | cos + | tan −
Coterminal Angles \(\theta_{coterminal} = \theta \pm 360°\) or \(\theta \pm 2\pi\)

Reference Angle Formulas

  • Quadrant I: Reference angle = \(\theta\)
  • Quadrant II: Reference angle = \(180° - \theta\) or \(\pi - \theta\)
  • Quadrant III: Reference angle = \(\theta - 180°\) or \(\theta - \pi\)
  • Quadrant IV: Reference angle = \(360° - \theta\) or \(2\pi - \theta\)
💡 ASTC Memory Aid

"All Students Take Calculus" — tells you which functions are positive in each quadrant: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4).

3 Trig Ratios in Right Triangles

The six trig functions relate an angle to the ratios of a right triangle's sides: opposite, adjacent, and hypotenuse.

Primary Trig Functions (SOH-CAH-TOA)

Sine
\(\sin\theta = \frac{\text{opp}}{\text{hyp}}\)
Cosine
\(\cos\theta = \frac{\text{adj}}{\text{hyp}}\)
Tangent
\(\tan\theta = \frac{\text{opp}}{\text{adj}}\)

Reciprocal Functions

Cosecant
\(\csc\theta = \frac{1}{\sin\theta}\)
Secant
\(\sec\theta = \frac{1}{\cos\theta}\)
Cotangent
\(\cot\theta = \frac{1}{\tan\theta}\)

4 The Unit Circle

The unit circle has radius 1 centered at the origin. For any angle \(\theta\), the point on the circle is \((\cos\theta, \sin\theta)\).

Unit Circle Definition Point \((x, y)\) on unit circle: \(x = \cos\theta\), \(y = \sin\theta\), \(\tan\theta = \frac{y}{x}\)
Pythagorean Identity \(\sin^2\theta + \cos^2\theta = 1\)

Additional Pythagorean Identities

\(1 + \tan^2\theta = \sec^2\theta\)
\(1 + \cot^2\theta = \csc^2\theta\)

5 Special Angles & Values

Memorize exact values for the most common angles: \(0°, 30°, 45°, 60°, 90°\) (or their radian equivalents).

Degrees Radians \(\sin\theta\) \(\cos\theta\) \(\tan\theta\)
\(0°\) \(0\) \(0\) \(1\) \(0\)
\(30°\) \(\frac{\pi}{6}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\)
\(45°\) \(\frac{\pi}{4}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) \(1\)
\(60°\) \(\frac{\pi}{3}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\)
\(90°\) \(\frac{\pi}{2}\) \(1\) \(0\) undefined
💡 Memory Trick

For sine values at 0°, 30°, 45°, 60°, 90°: think \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\). For cosine, reverse the order!

6 Inverse Trigonometric Functions

Inverse trig functions find the angle when given a ratio. They are restricted to specific domains to ensure one output per input.

Inverse Sine (arcsin)
\(y = \sin^{-1}(x) \Leftrightarrow x = \sin y\)
Domain: \([-1, 1]\) → Range: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
Inverse Cosine (arccos)
\(y = \cos^{-1}(x) \Leftrightarrow x = \cos y\)
Domain: \([-1, 1]\) → Range: \([0, \pi]\)
Inverse Tangent (arctan)
\(y = \tan^{-1}(x) \Leftrightarrow x = \tan y\)
Domain: \((-\infty, \infty)\) → Range: \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
📌 Example

Find: \(\sin^{-1}\left(\frac{1}{2}\right)\)

Solution: We need the angle where \(\sin\theta = \frac{1}{2}\) in range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Answer: \(\frac{\pi}{6}\) (or \(30°\))

7 Solving Right Triangles & Trig Equations

Use trig ratios to find unknown sides and angles in right triangles. For equations, isolate the trig function first.

Solving Right Triangles

  • Given angle \(A\) and hypotenuse \(c\): \(a = c \cdot \sin A\), \(b = c \cdot \cos A\)
  • Given angle \(A\) and adjacent side \(b\): \(a = b \cdot \tan A\)
  • Given two sides: Use inverse trig to find angle

Solving Trig Equations

  1. Isolate the trig function
  2. Use inverse function to find reference angle
  3. Find ALL solutions in given interval using quadrant signs
  4. Add period multiples if finding general solution
📌 Example

Solve: \(2\sin\theta - 1 = 0\) for \(\theta \in [0, 2\pi)\)

Isolate: \(\sin\theta = \frac{1}{2}\)

Reference angle: \(\frac{\pi}{6}\)

Solutions: \(\theta = \frac{\pi}{6}\) (Q1) and \(\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\) (Q2)

8 Law of Sines & Law of Cosines

These laws solve oblique triangles (non-right triangles). Use when you don't have a right angle.

Law of Sines
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Use when: AAS, ASA, or SSA (ambiguous case)
Law of Cosines
\(c^2 = a^2 + b^2 - 2ab\cos C\)
Use when: SAS or SSS
⚠️ Ambiguous Case (SSA)

When given two sides and an angle opposite one of them, there may be 0, 1, or 2 possible triangles. Check if \(\sin B > 1\) (no solution) or if the angle could be in two quadrants.

9 Triangle Area Formulas

Calculate the area of any triangle using these formulas based on what information you have.

Sine Formula
\(A = \frac{1}{2}ab\sin C\)
Use when: Two sides and included angle (SAS)
Heron's Formula
\(A = \sqrt{s(s-a)(s-b)(s-c)}\)
where \(s = \frac{a+b+c}{2}\)
Use when: Three sides known (SSS)
📌 Example: Heron's Formula

Find area: Triangle with sides \(a = 5\), \(b = 6\), \(c = 7\)

Semi-perimeter: \(s = \frac{5+6+7}{2} = 9\)

Area: \(A = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} = 6\sqrt{6}\)

📋 Quick Reference

Degrees ↔ Radians

\(\times \frac{\pi}{180}\) or \(\times \frac{180}{\pi}\)

Arc Length

\(s = r\theta\)

SOH-CAH-TOA

sin=O/H, cos=A/H, tan=O/A

Pythagorean ID

\(\sin^2 + \cos^2 = 1\)

Law of Sines

\(\frac{a}{\sin A} = \frac{b}{\sin B}\)

Law of Cosines

\(c^2 = a^2 + b^2 - 2ab\cos C\)

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