🎯 Foundation & Unit Circle
📐 Trigonometry Master Reference
🎯 Foundation & Unit Circle
30° (π/6)
(√3/2, 1/2)
(√3/2, 1/2)
45° (π/4)
(√2/2, √2/2)
(√2/2, √2/2)
60° (π/3)
(1/2, √3/2)
(1/2, √3/2)
90° (π/2)
(0, 1)
(0, 1)
120° (2π/3)
(-1/2, √3/2)
(-1/2, √3/2)
150° (5π/6)
(-√3/2, 1/2)
(-√3/2, 1/2)
180° (π)
(-1, 0)
(-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
- Isolate trig function
- Find reference angle
- Determine quadrants from sign
- Apply: θ = ref angle, π - ref, π + ref, 2π - ref
- Add + 2πn for general solution
📐 Right Triangle Problems
- Label sides (Opp, Adj, Hyp)
- Choose appropriate function (SOH-CAH-TOA)
- Set up equation
- Solve for unknown
- Check if answer is reasonable
🌀 Simplifying Expressions
- Convert everything to sin/cos
- Use Pythagorean identities
- Factor when possible
- Look for conjugate multiplication
- Check with sample value