Graphing Trigonometric Functions

📐 Sine & Cosine Graphs

Part 1 of 7 — Sine & Cosine Graphs

$y = A\sin(Bx + C) + D$

• A: amplitude (vertical stretch)
• B: affects period ($T = \frac{2\pi}{|B|}$)
• C: phase shift
• D: vertical shift (midline)

Sine starts at midline; cosine starts at maximum.

Worked Example

$y = \sin(x)$. Key features?

Amplitude = 1, Period = 2π, Midline: y = 0 ✅

Concept Check 🎯

Amplitude 🧮

1. Amplitude of $\sin(x)$?

2. Amplitude of $\cos(x)$?

3. Amplitude of $2\sin(x)$?

Concept Check 🔍

Practice

Challenge Question 📋

📊 Amplitude & Period

Part 2 of 7 — Amplitude & Period

• Amplitude = $|A|$ (distance from midline to max/min)
• Period = $\frac{2\pi}{|B|}$ (one full cycle length)

For : amplitude = 3, period =

🔢 Phase Shift

Part 3 of 7 — Phase Shift

$y = A\sin(B(x - h)) + D$

Phase shift = $h = -\frac{C}{B}$ (horizontal shift)

📈 Modeling with Sinusoids

Part 4 of 7 — Modeling with Sinusoids

Many real-world phenomena are periodic:

• Temperature over a year
• Tides, daylight hours
• Sound waves, pendulums

To model: find amplitude (half of max−min), midline (average of max and min), and period (time for one cycle).

Worked Example

Temperature: max 90°F, min 30°F, period 12 months.

$A = \frac{90-30}{2} = 30$,

🧮 Inverse Trig Functions

Part 5 of 7 — Inverse Trig Functions

• $\sin^{-1}(x)$ or $\arcsin(x)$: returns angle whose sine is $x$; range $$

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Apply all trig modeling concepts:

• Read amplitude, period, shifts from a graph or description
• Write a sinusoidal model
• Evaluate inverse trig expressions

Worked Example

Ferris wheel: radius 20 ft, center 25 ft high, period 60 sec.

$h(t) = -20\cos\left(\frac{2\pi}{60}t\right) + 25 = -20\cos\left(\frac{\pi}{30}t\right) + 25$ ✅

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $y = A\sin(B(x-h)) + D$
• Amplitude = $|A|$