📐 Trigonometry Master Reference

Unit Circle Identities Formulas Graphs Solver Problems

🎯 Foundation & Unit Circle

30° (π/6)
(√3/2, 1/2)
45° (π/4)
(√2/2, √2/2)
60° (π/3)
(1/2, √3/2)
90° (π/2)
(0, 1)
120° (2π/3)
(-1/2, √3/2)
150° (5π/6)
(-√3/2, 1/2)
180° (π)
(-1, 0)

🔺 Right Triangle Definitions

SOH-CAH-TOA Mnemonic:

Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

📏 Reciprocal Functions

  • csc θ = 1/sin θ = Hypotenuse/Opposite
  • sec θ = 1/cos θ = Hypotenuse/Adjacent
  • cot θ = 1/tan θ = Adjacent/Opposite

🧮 Essential Identities

⚖️ Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

🔄 Angle Sum & Difference

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

⚡ Double & Half Angle

sin(2θ) = 2 sinθ cosθ

cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

sin(θ/2) = ±√[(1 - cosθ)/2]

cos(θ/2) = ±√[(1 + cosθ)/2]

🔀 Product-to-Sum

sinA cosB = ½[sin(A+B) + sin(A-B)]

cosA cosB = ½[cos(A+B) + cos(A-B)]

📈 Function Graphs & Laws

📊 Basic Graphs Pattern

y = sin x: Starts at 0, max = 1, period = 2π
y = cos x: Starts at 1, max = 1, period = 2π
y = tan x: Vertical asymptotes at π/2 + nπ

General Form: y = A sin(Bx - C) + D

  • A = Amplitude (vertical stretch)
  • B affects Period: Period = 2π/|B|
  • C = Phase Shift (horizontal shift)
  • D = Vertical Shift

⚖️ Law of Sines & Cosines

Law of Sines: a/sinA = b/sinB = c/sinC

Law of Cosines:
a² = b² + c² - 2bc cosA
cosA = (b² + c² - a²)/(2bc)

🎯 Inverse Functions Range

  • sin⁻¹(x): [-π/2, π/2]
  • cos⁻¹(x): [0, π]
  • tan⁻¹(x): (-π/2, π/2)

🛠️ Trig Problem Solver

🔢 Value Calculator

🔄 Identity Verifier

🎯 Practice Problem Generator

🎯 Problem-Solving Strategies

🔍 Solving Trig Equations

  1. Isolate trig function
  2. Find reference angle
  3. Determine quadrants from sign
  4. Apply: θ = ref angle, π - ref, π + ref, 2π - ref
  5. Add + 2πn for general solution

📐 Right Triangle Problems

  1. Label sides (Opp, Adj, Hyp)
  2. Choose appropriate function (SOH-CAH-TOA)
  3. Set up equation
  4. Solve for unknown
  5. Check if answer is reasonable

🌀 Simplifying Expressions

  1. Convert everything to sin/cos
  2. Use Pythagorean identities
  3. Factor when possible
  4. Look for conjugate multiplication
  5. Check with sample value

💡 Memory Aids & Quick Tips

ASTC: All Students Take Calculus (Quadrants where functions are positive: I-All, II-Sin, III-Tan, IV-Cos)

Special Angles: sin 0°,30°,45°,60°,90° = √0/2, √1/2, √2/2, √3/2, √4/2

Cast Rule: Cosine All Sine Tangent (for sign determination)

⏱️ Recommended: Practice 15 min daily |