Basic Trigonometric Functions

When you extend the unit circle idea and let the angle θ keep increasing (or decreasing), the sine and cosine values repeat in a beautiful, wave-like pattern. These are the trigonometric functions -- mathematical models for anything that cycles: sound waves, tides, heartbeats, seasons, alternating current.

Sine and Cosine as Functions

The functions y = sin(x) and y = cos(x) have these key properties:

Property	sin(x)	cos(x)
Domain	All real numbers	All real numbers
Range	[-1, 1]	[-1, 1]
Period	2π	2π
Amplitude	1	1
y-intercept	0	1
Symmetry	Odd: sin(-x) = -sin(x)	Even: cos(-x) = cos(x)

The Tangent Function

tan(x) = sin(x)/cos(x). It is undefined wherever cos(x) = 0 (at x = π/2 + nπ).

Property	tan(x)
Domain	All real numbers except x = π/2 + nπ
Range	All real numbers (-∞, +∞)
Period	π
Vertical asymptotes	At x = π/2 + nπ

Transformations: Amplitude, Period, Phase Shift

The general sinusoidal function is:

y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D

Parameter	Effect
A (amplitude)	Vertical stretch. The wave oscillates between D - \|A\| and D + \|A\|.
B	Affects period: period = 2π/\|B\|. Larger B means faster oscillation.
C (phase shift)	Horizontal shift. Positive C shifts the graph right.
D (vertical shift)	Moves the midline up (positive D) or down (negative D).

Worked Example 1 -- Reading a Graph

Identify the amplitude, period, and midline of y = 3 sin(2x) + 1.

Amplitude = |A| = 3. The wave extends 3 units above and below the midline.
Period = 2π/|B| = 2π/2 = π. One full cycle completes every π units.
Midline: y = D = 1. The wave oscillates between 1 - 3 = -2 and 1 + 3 = 4.

Worked Example 2 -- Writing an Equation

A wave has amplitude 5, period 4π, and is shifted up by 2. Write its equation using cosine.

A = 5, D = 2.
Period = 2π/B = 4π, so B = 2π/(4π) = 1/2.
No phase shift mentioned, so C = 0. Equation: y = 5 cos(x/2) + 2.

Worked Example 3 -- Phase Shift

Describe the transformation from y = sin(x) to y = sin(x - π/4).

This is a phase shift of π/4 to the right. Every feature of the sine curve (peaks, zeros, troughs) moves π/4 units to the right.

Connecting Sine and Cosine

Cosine is just sine shifted left by π/2: cos(x) = sin(x + π/2). They are the same wave at different starting points. When modeling real-world cycles, pick whichever makes your equation simpler.

Common Mistake

Confusing the "B" value with the period. The period is 2π/divided by B, not multiplied. If B = 3, the period is 2π/3 (shorter, faster oscillation), not 6π.

Practice Problems

State the amplitude, period, and midline of y = -2 cos(3x) - 4.
Show Solution

Amplitude = |-2| = 2. Period = 2π/3. Midline: y = -4. Range: [-6, -2].
Write the equation of a sine function with amplitude 4, period π, and no vertical shift.
Show Solution

B = 2π/π = 2. Equation: y = 4 sin(2x).
What is the period of y = tan(x/2)?
Show Solution

Period of tangent = π/|B| = π/(1/2) = 2π.
Sketch or describe one full cycle of y = sin(x) starting at x = 0. Where are the zeros, maximum, and minimum?
Show Solution

Zeros at x = 0, π, 2π. Maximum of 1 at x = π/2. Minimum of -1 at x = 3π/2.
The function y = 3 sin(2(x - π/6)) + 1 has what phase shift?
Show Solution

Phase shift = C = π/6 to the right.

Summary

sin(x) and cos(x) are periodic functions with period 2π and range [-1, 1].
tan(x) has period π and vertical asymptotes where cos(x) = 0.
The general form y = A sin(B(x - C)) + D controls amplitude, period, phase shift, and vertical shift.
Period = 2π/|B| for sine and cosine; π/|B| for tangent.