Trigonometric Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. General Form of Sine and Cosine Functions
Standard Forms:
\[ y = A\sin(B(x - C)) + D \] \[ y = A\cos(B(x - C)) + D \]
Parameters:
• A = Amplitude (vertical stretch/compression)
• B = Affects the period
• C = Phase shift (horizontal shift)
• D = Vertical shift (midline)
2. Properties of Sine and Cosine Functions
Amplitude:
\[ \text{Amplitude} = |A| \]
• Distance from midline to maximum or minimum
• Always positive
Period:
\[ \text{Period} = \frac{2\pi}{|B|} \]
• Horizontal length of one complete cycle
• For standard sin/cos: Period = \( 2\pi \)
Phase Shift (Horizontal Shift):
\[ \text{Phase Shift} = C \]
• If C > 0: Shift RIGHT
• If C < 0: Shift LEFT
Midline (Vertical Shift):
\[ \text{Midline: } y = D \]
• Horizontal line about which the function oscillates
• Average of maximum and minimum values
Maximum and Minimum Values:
\[ \text{Maximum} = D + |A| \] \[ \text{Minimum} = D - |A| \]
3. Write Equations from Graphs
Steps:
1. Determine A (Amplitude):
\( A = \frac{\text{max} - \text{min}}{2} \)
2. Determine D (Midline):
\( D = \frac{\text{max} + \text{min}}{2} \)
3. Determine B (from Period):
\( B = \frac{2\pi}{\text{Period}} \)
4. Decide: Sine or Cosine?
• Cosine: Starts at maximum or minimum
• Sine: Starts at midline
5. Determine C (Phase Shift):
Horizontal distance from expected starting point
Example:
Given: Graph with max = 5, min = -1, period = \( \pi \)
Amplitude: \( A = \frac{5-(-1)}{2} = 3 \)
Midline: \( D = \frac{5+(-1)}{2} = 2 \)
Period = \( \pi \), so \( B = \frac{2\pi}{\pi} = 2 \)
Equation: \( y = 3\sin(2x) + 2 \) or \( y = 3\cos(2x) + 2 \)
4. Graph Sine Functions
Parent Function \( y = \sin(x) \):
• Amplitude: 1
• Period: \( 2\pi \)
• Midline: y = 0
• Starts at (0, 0)
• Maximum at \( x = \frac{\pi}{2} \), Minimum at \( x = \frac{3\pi}{2} \)
Key Points for One Period:
| x | y = sin(x) | Point |
|---|---|---|
| 0 | 0 | Start at midline |
| \( \frac{\pi}{2} \) | 1 | Maximum |
| \( \pi \) | 0 | Return to midline |
| \( \frac{3\pi}{2} \) | -1 | Minimum |
| \( 2\pi \) | 0 | Complete cycle |
Steps to Graph:
1. Identify amplitude, period, phase shift, and midline
2. Draw the midline \( y = D \)
3. Mark maximum and minimum values
4. Divide period into 4 equal parts
5. Plot key points and draw smooth curve
5. Graph Cosine Functions
Parent Function \( y = \cos(x) \):
• Amplitude: 1
• Period: \( 2\pi \)
• Midline: y = 0
• Starts at (0, 1) - Maximum
• Minimum at \( x = \pi \)
Key Points for One Period:
| x | y = cos(x) | Point |
|---|---|---|
| 0 | 1 | Start at maximum |
| \( \frac{\pi}{2} \) | 0 | Crosses midline |
| \( \pi \) | -1 | Minimum |
| \( \frac{3\pi}{2} \) | 0 | Return to midline |
| \( 2\pi \) | 1 | Complete cycle |
Relationship Between Sine and Cosine:
\[ \cos(x) = \sin\left(x + \frac{\pi}{2}\right) \]
Cosine is sine shifted left by \( \frac{\pi}{2} \)
6. Transformations and Translations
Types of Transformations:
Vertical Stretch/Compression (|A|):
• |A| > 1: Vertical stretch (taller)
• 0 < |A| < 1: Vertical compression (shorter)
• A < 0: Reflection over x-axis
Horizontal Stretch/Compression (B):
• B > 1: Horizontal compression (more cycles)
• 0 < B < 1: Horizontal stretch (fewer cycles)
Phase Shift (C):
• C > 0: Shift right
• C < 0: Shift left
Vertical Shift (D):
• D > 0: Shift up
• D < 0: Shift down
7. Complete Examples
Example 1: Sine Function
Find properties: \( y = 3\sin(2x - \pi) + 1 \)
Amplitude: \( |A| = |3| = 3 \)
Period: \( \frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi \)
Phase Shift: \( C = \frac{\pi}{2} \) (right)
Midline: \( y = 1 \)
Maximum: \( 1 + 3 = 4 \)
Minimum: \( 1 - 3 = -2 \)
Example 2: Cosine Function
Find properties: \( y = -2\cos\left(\frac{1}{2}x + \pi\right) - 3 \)
Amplitude: \( |-2| = 2 \)
Period: \( \frac{2\pi}{1/2} = 4\pi \)
Phase Shift: \( -2\pi \) (left)
Midline: \( y = -3 \)
Reflected over x-axis (A is negative)
Maximum: \( -3 + 2 = -1 \)
Minimum: \( -3 - 2 = -5 \)
Example 3: Write from Graph
Given: Maximum = 7, Minimum = 1, Period = \( 4\pi \), starts at midline going up
Amplitude: \( A = \frac{7-1}{2} = 3 \)
Midline: \( D = \frac{7+1}{2} = 4 \)
Period = \( 4\pi \), so \( B = \frac{2\pi}{4\pi} = \frac{1}{2} \)
Starts at midline going up → Use sine
Equation: \( y = 3\sin\left(\frac{1}{2}x\right) + 4 \)
8. Quick Reference Summary
Key Formulas:
General Form: \( y = A\sin(B(x - C)) + D \) or \( y = A\cos(B(x - C)) + D \)
Amplitude: \( |A| \)
Period: \( \frac{2\pi}{|B|} \)
Phase Shift: C
Midline: \( y = D \)
Maximum: \( D + |A| \)
Minimum: \( D - |A| \)
Key Differences:
• Sine: Starts at midline, goes up first
• Cosine: Starts at maximum or minimum
📚 Study Tips
✓ Amplitude is always positive (use |A|)
✓ Period = 2π/|B| - larger B means shorter period
✓ Sine starts at midline; cosine starts at max/min
✓ Midline = (max + min)/2; Amplitude = (max - min)/2
✓ Negative A reflects graph over x-axis