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 $y = 3\sin(2x)$: amplitude = 3, period = $\frac{2\pi}{2} = \pi$

Worked Example

$y = 4\cos(3x)$. Amplitude? Period?

Amplitude = 4, Period = $\frac{2\pi}{3} \approx 2.09$ ✅

Concept Check 🎯

Find Amplitude 🧮

1. $3\sin(2x)$: Amplitude?

2. $5\cos(x)$: Amplitude?

3. $\sin(4x)$: Amplitude?

Concept Check 🔍

Practice

Challenge Question 📋

🔢 Phase Shift

Part 3 of 7 — Phase Shift

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

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

• Shift right when $h > 0$
• Shift left when $h < 0$

Vertical shift = $D$ (moves the midline)

Worked Example

$y = \sin(x - \frac{\pi}{4})$. Phase shift?

Shift right by $\frac{\pi}{4}$ ✅

Concept Check 🎯

Shifts 🧮

1. $\sin(x) + 3$. Midline at $y =$ ?

2. $\sin(x - \pi)$. Midline at $y =$ ?

3. $\cos(x) + 2$. Midline at $y =$ ?

Concept Check 🔍

Practice

Challenge Question 📋

📈 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.

$A = \frac{90-30}{2} = 30$, $D = \frac{90+30}{2} = 60$ $B = \frac{2\pi}{12} = \frac{\pi}{6}$

$T(t) = 30\sin\left(\frac{\pi}{6}t\right) + 60$ ✅

Concept Check 🎯

Find Amplitude 🧮

Amplitude = (max − min) / 2

1. Max=80, min=40. Amplitude?

2. Max=100, min=20. Amplitude?

3. Max=50, min=10. Amplitude?

Concept Check 🔍

Practice

Challenge Question 📋

🧮 Inverse Trig Functions

Part 5 of 7 — Inverse Trig Functions

Sine, cosine and tangent are periodic, so each output comes from infinitely many angles. To build an inverse, the domain of each function is restricted to one interval where it takes every output exactly once. That interval becomes the range of the inverse function, and AP Precalculus measures it in radians.

🔑 Key Concept: $\arcsin(x)$ is the angle in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ whose sine is $x$. A negative input gives a negative angle for arcsine and arctangent, but an angle between $\frac{\pi}{2}$ and $\pi$ for arccosine.

Worked Examples

1) $\arcsin\left(-\frac{\sqrt{2}}{2}\right)$. The reference angle with sine $\frac{\sqrt{2}}{2}$ is $\frac{\pi}{4}$. The input is negative and the range is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, so the answer is in Quadrant IV: $-\frac{\pi}{4}$.

2) $\arccos\left(-\frac{1}{2}\right)$. The reference angle with cosine $\frac{1}{2}$ is $\frac{\pi}{3}$. The range $[0, \pi]$ puts a negative cosine in Quadrant II: $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$. The angle $\frac{4\pi}{3}$ also has cosine $-\frac{1}{2}$, but it is outside the range.

3) $\arctan(-\sqrt{3})$. The reference angle with tangent $\sqrt{3}$ is $\frac{\pi}{3}$. A negative input gives a Quadrant IV angle: $-\frac{\pi}{3}$.

Concept Check 🎯

Inverse Trig in Radians 🧮

Give each answer in radians as a fraction of pi, such as "pi/3", "-pi/4" or "0".

1. $\arccos(1) =$ ?

2. $\arctan(-1) =$ ?

3. $\arccos\left(-\frac{\sqrt{3}}{2}\right) =$ ?

Ranges in Radians 🔍

Using an Inverse to Solve an Equation

An inverse function gives one solution. The other solutions come from the unit circle and the period.

Solve $\sin x = 0.4$ for $0 \le x < 2\pi$.

In a model such as $h(t) = 10 + 4\sin\left(\frac{\pi}{6}t\right)$, the same idea finds every time $t$ when $h(t)$ reaches a given value: solve for the sine, apply the inverse, then use symmetry and the period to list the other times.

Challenge Question 📋

🛠️ 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.

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

(Negative cosine starts at the bottom)

Concept Check 🎯

Applications 🧮

Max = 70, min = 30:

1. Midline?

2. Amplitude?

3. Amplitude of $5\sin(\pi x) + 2$?

Concept Check 🔍

Practice

Challenge Question 📋

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $y = A\sin(B(x-h)) + D$
• Amplitude = $|A|$
• Period = $2\pi / |B|$
• Phase shift = $h$
• Midline = $D$
• Amplitude = (max−min)/2
• Midline = (max+min)/2

Worked Example

$y = 3\sin(2x) + 1$

A=3, B=2, D=1. Amplitude=3, Period=π, Midline: y=1 ✅

Concept Check 🎯

Review 🧮

1. Amplitude of $4\cos(x)$?

2. $\sin^{-1}(1)$ in radians? (Enter like "pi/3")

3. Midline of $\sin(x) + 1$?

Concept Check 🔍

Practice

Challenge Question 📋