📌 Introduction

Sine and cosine functions are periodic functions that repeat their values in regular intervals. They are fundamental to understanding oscillatory motion, waves, and circular motion.

Standard Forms:

The Four Key Properties:

Parent Functions (Basic Forms):

📝 Example - Finding Properties:

Find the amplitude, period, phase shift, and vertical shift of \( y = 3\sin(2x - \pi) + 1 \)

Compare with \( y = A\sin(Bx - C) + D \):
\( A = 3, B = 2, C = \pi, D = 1 \)

Amplitude: \( |A| = |3| = 3 \)
Period: \( \frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi \)
Phase Shift: \( \frac{C}{B} = \frac{\pi}{2} \) (right)
Vertical Shift: \( D = 1 \) (up 1)
Midline: \( y = 1 \)
Range: \( [-2, 4] \) (from \( 1-3 \) to \( 1+3 \))

Step-by-Step Process:

1. Determine if it's sine or cosine
   • Sine: starts at midline, goes up (or down if reflected)
   • Cosine: starts at maximum or minimum
2. Find the amplitude: \( A = \frac{\text{max} - \text{min}}{2} \)
3. Find the vertical shift (midline): \( D = \frac{\text{max} + \text{min}}{2} \)
4. Find the period: Measure one complete cycle
5. Calculate B: \( B = \frac{2\pi}{\text{period}} \)
6. Find the phase shift: How far the graph is shifted horizontally
7. Calculate C: \( C = B \times \text{phase shift} \)
8. Check for reflection: If graph is flipped, \( A \) is negative

📝 Example - Writing Equation from Graph:

A sine graph has a maximum of 5, minimum of -1, period of 4π, and starts at the midline at \( x = 0 \). Write the equation.

Step 1: It's a sine function (starts at midline)

Step 2: Amplitude:
\( A = \frac{5 - (-1)}{2} = \frac{6}{2} = 3 \)

Step 3: Vertical shift:
\( D = \frac{5 + (-1)}{2} = \frac{4}{2} = 2 \)

Step 4: Period = \( 4\pi \)

Step 5: Find B:
\( B = \frac{2\pi}{4\pi} = \frac{1}{2} \)

Step 6: No phase shift (\( C = 0 \))

Equation: \( y = 3\sin\left(\frac{1}{2}x\right) + 2 \)

Key Points for Basic Sine Function \( y = \sin x \):

Key Points for Basic Cosine Function \( y = \cos x \):

Steps to Graph Transformed Functions:

1. Identify all parameters: A, B, C, D
2. Draw the midline at \( y = D \)
3. Mark maximum and minimum at \( D + |A| \) and \( D - |A| \)
4. Determine the period and mark one complete cycle
5. Apply phase shift (horizontal translation)
6. Plot 5 key points within one period
7. Sketch smooth curve through the points
8. Extend the pattern if needed

Effects of Each Parameter:

📝 Complete Example - Graphing:

Graph \( y = -2\cos(3x + \pi) - 1 \)

Step 1: Rewrite in standard form:
\( y = -2\cos\left(3\left(x + \frac{\pi}{3}\right)\right) - 1 \)

Step 2: Identify parameters:
\( A = -2 \) (reflected, amplitude = 2)
\( B = 3 \)
\( C = -\pi \) (or phase shift = \( -\frac{\pi}{3} \), left)
\( D = -1 \)

Step 3: Find properties:
Amplitude: 2
Period: \( \frac{2\pi}{3} \)
Phase shift: \( \frac{\pi}{3} \) left
Midline: \( y = -1 \)
Maximum: \( -1 + 2 = 1 \)
Minimum: \( -1 - 2 = -3 \)
Reflected (starts at minimum instead of maximum)

Step 4: Plot key points and sketch

Key Relationships:

⚡ Quick Summary

• General form: \( y = A\sin(Bx - C) + D \) or \( y = A\cos(Bx - C) + D \)
• Amplitude tells how tall the waves are
• Period tells how long one complete cycle is
• Phase shift moves the graph horizontally
• Vertical shift moves the midline up or down
• Range: \( [D - |A|, D + |A|] \)

📚 Sine vs Cosine - Quick Comparison

⚠️ Common Mistakes to Avoid