Trigonometric Functions

Trigonometric functions extend the chapter-02-relations-and-functions concept to angles and periodic phenomena. Originally developed to solve triangle problems, trigonometry now describes oscillations, waves, and circular motion. A periodic function repeats its pattern over regular intervals—just like the seasons cycle yearly or a pendulum swings back and forth. This chapter introduces sine, cosine, tangent, and their companions, which model everything from sound waves to planetary orbits. Mastery of trigonometric functions opens the door to understanding calculus and physics.

From Triangles to Circles

The journey to trigonometric functions begins with right triangles. In a right triangle, we defined sine, cosine, and tangent using ratios of sides. But these definitions only work for angles between 0° and 90°. What about angles of 120° or 200°?

The answer: use a circle. Place a circle of radius 1 (called the unit circle) at the origin of a coordinate system. Now, imagine a ray starting from the center, making an angle θ with the positive x-axis. This ray intersects the circle at a point. That point's coordinates are (cos θ, sin θ).

This elegant definition extends sine and cosine to any angle, positive or negative, small or huge. The definitions are:

cos θ = x-coordinate of the point on the unit circle

sin θ = y-coordinate of the point on the unit circle

Angle Measurement

Angles can be measured in degrees (where a full rotation is 360°) or radians (where a full rotation is 2π ≈ 6.28). Radians are the natural unit for mathematics because many formulas simplify.

Conversion: θ (radians) = θ (degrees) × π/180

Key Trigonometric Functions

Sine (sin θ): The y-coordinate. It oscillates between -1 and 1. At θ = 0, sin(0) = 0. At θ = π/2 (90°), sin(π/2) = 1.

Cosine (cos θ): The x-coordinate. Also oscillates between -1 and 1. At θ = 0, cos(0) = 1. At θ = π/2, cos(π/2) = 0.

Tangent (tan θ) = sin θ / cos θ: The slope of the ray. Can be any real number. Undefined where cos θ = 0 (at π/2, 3π/2, etc.).

Reciprocal functions:

• Cosecant (csc θ) = 1/sin θ
• Secant (sec θ) = 1/cos θ
• Cotangent (cot θ) = 1/tan θ

Periodicity: The Heart of Trigonometry

Both sine and cosine repeat every 2π radians (360°):

sin(θ + 2π) = sin θ

cos(θ + 2π) = cos θ

Tangent repeats every π radians:

tan(θ + π) = tan θ

This periodic behavior makes trigonometric functions perfect for describing anything cyclical.

Essential Identities

These relationships hold for any angle and are invaluable for simplifying expressions:

Pythagorean Identity: sin²θ + cos²θ = 1 (derived directly from the unit circle—x² + y² = 1)

Angle Sum Formulas:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B - sin A sin B

Double Angle Formulas:

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

Graphs of Trigonometric Functions

y = sin x: A smooth wave oscillating between -1 and 1, crossing zero at x = 0, π, 2π, etc.

y = cos x: Similar to sine but shifted left by π/2. At x = 0, it starts at 1.

y = tan x: Has vertical asymptotes (breaks) where cos x = 0. Between asymptotes, it increases from -∞ to +∞.

Transformations

Just as you can shift and stretch other functions, you can transform trigonometric ones:

y = A sin(Bx + C) + D

• A changes the amplitude (vertical stretch)
• B changes the period (horizontal stretch)—the period becomes 2π/B
• C causes a horizontal shift
• D causes a vertical shift

Real-World Applications

Sound waves are sine waves. The height of a Ferris wheel seat as a function of time follows cosine. AC electric current varies sinusoidally. Radio waves, light, and all electromagnetic radiation involve sine and cosine. Navigators use trigonometry. Architects and engineers use it for construction and design.

Socratic Questions

1. Why is the unit circle (radius 1) chosen for defining trigonometric functions? What would change if we used a circle with radius 2? What principle remains the same?

1. Sine and cosine are periodic with period 2π. Can you explain why they return to the same value after a full rotation? What does periodicity tell us about the underlying symmetry?

1. The Pythagorean identity says sin²θ + cos²θ = 1. Geometrically, what property of the unit circle does this represent? Can you visualize this identity?

1. Given that tan(θ) = sin(θ)/cos(θ), why is tangent undefined at θ = π/2 and 3π/2? What happens to the tangent line at these angles visually?

1. If you're modeling a real-world oscillating phenomenon with y = A sin(Bx + C) + D, which parameter (A, B, C, or D) represents the "speed" of oscillation? How would you determine these parameters from experimental data?