Learning Objectives
- Understand angles, their measurement in degrees and radians
- Learn trigonometric functions and their properties
- Study trigonometric identities and equations
- Understand graphs of trigonometric functions
Key Concepts
Angles and Measurement
Degree measure: 1 full rotation = 360°. Radian measure: 1 full rotation = 2π radians. Conversion: π radians = 180°. So 1° = π/180 rad and 1 rad = 180°/π ≈ 57.3°. Arc length: l = rθ (θ in radians). Area of sector: A = (1/2)r2θ.
Trigonometric Functions
For angle θ in standard position with point P(x, y) on unit circle: sin θ = y, cos θ = x, tan θ = y/x, csc θ = 1/y, sec θ = 1/x, cot θ = x/y.
Signs in quadrants (ASTC rule): All positive (Q1), Sin positive (Q2), Tan positive (Q3), Cos positive (Q4). Memory aid: "All Students Take Coffee".
Domain and Range: sin θ, cos θ: Domain = R, Range = [-1, 1]. tan θ: Domain = R - {(2n+1)π/2}, Range = R. sec θ: Range = (-∞, -1] ∪ [1, ∞). csc θ: Range = (-∞, -1] ∪ [1, ∞).
Fundamental Identities
Pythagorean: sin2θ + cos2θ = 1; 1 + tan2θ = sec2θ; 1 + cot2θ = csc2θ.
Compound Angles: sin(A ± B) = sin A cos B ± cos A sin B. cos(A ± B) = cos A cos B ∓ sin A sin B. tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
Double Angle: sin 2A = 2 sin A cos A. cos 2A = cos2A - sin2A = 2cos2A - 1 = 1 - 2sin2A. tan 2A = 2tan A/(1 - tan2A).
Triple Angle: sin 3A = 3sin A - 4sin3A. cos 3A = 4cos3A - 3cos A.
Sum to Product: sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2). sin C - sin D = 2 cos((C+D)/2) sin((C-D)/2). cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2). cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2).
Product to Sum: 2 sin A cos B = sin(A+B) + sin(A-B). 2 cos A cos B = cos(A-B) + cos(A+B). 2 sin A sin B = cos(A-B) - cos(A+B).
Trigonometric Equations
General solutions: sin θ = sin α ⟹ θ = nπ + (-1)nα. cos θ = cos α ⟹ θ = 2nπ ± α. tan θ = tan α ⟹ θ = nπ + α. Where n ∈ Z (integers). Principal value: Solution in the principal range.
Summary
Trigonometric functions are defined using the unit circle. Radian measure relates arc length to radius. Key identities include Pythagorean, compound angle, double angle, and sum-to-product formulas. General solutions of trigonometric equations involve integral multiples of π.
Important Terms
- Radian: Angle subtended by arc equal to radius; π rad = 180°
- Unit circle: Circle with radius 1 centered at origin
- Period: Smallest positive value T such that f(x + T) = f(x)
- Amplitude: Maximum displacement from the mean position
- Principal value: Unique solution of trig equation in a defined interval
- Compound angle: Angle expressed as sum or difference of two angles
Quick Revision
- π rad = 180°; 1 rad ≈ 57.3°
- sin2θ + cos2θ = 1 (most fundamental identity)
- ASTC: Q1 all+, Q2 sin+, Q3 tan+, Q4 cos+
- sin(A+B) = sinAcosB + cosAsinB
- cos2A = 1 - 2sin2A = 2cos2A - 1
- Period: sin, cos = 2π; tan, cot = π
- sin θ = 0 ⟹ θ = nπ; cos θ = 0 ⟹ θ = (2n+1)π/2
- General solution of sinθ = sinα: θ = nπ + (-1)nα