Grade 12 Graphing Sine and Cosine Functions Worksheets 2024 - Free Printable

Absolutely! Let’s solve each of the four problems step by step. For each, we’ll find the amplitude and period, then describe how to graph the function using radians.

---

## 🔷 General Form Reminder:

For functions of the form:
- `y = A·cos(Bθ)` or `y = A·sin(Bθ)`

→ Amplitude = |A|
→ Period = `2π / |B|`

---

## ✔ Problem 1: `y = 3 cos(2θ)`

➤ Amplitude:

A = 3 → Amplitude = |3| = 3

➤ Period:

B = 2 → Period = 2π / 2 = π

➤ Graphing Instructions:

- The cosine wave starts at its maximum (since no phase shift).
- Key points over one period (from θ = 0 to θ = π):
- At θ = 0 → y = 3·cos(0) = 3
- At θ = π/4 → y = 3·cos(π/2) = 0
- At θ = π/2 → y = 3·cos(π) = -3
- At θ = 3π/4 → y = 3·cos(3π/2) = 0
- At θ = π → y = 3·cos(2π) = 3

So, plot these 5 key points and connect them smoothly in a cosine wave shape.

✔ Answer for #1:
Amplitude = 3, Period = π

---

## ✔ Problem 2: `y = cos(3θ)`

➤ Amplitude:

A = 1 → Amplitude = |1| = 1

➤ Period:

B = 3 → Period = 2π / 3

➤ Graphing Instructions:

- Starts at max (cosine), so at θ=0, y=1.
- One full cycle from θ = 0 to θ = 2π/3.
- Divide the period into 4 equal parts for key points:
- θ = 0 → y = cos(0) = 1
- θ = (2π/3)/4 = π/6 → y = cos(π/2) = 0
- θ = 2π/6 = π/3 → y = cos(π) = -1
- θ = π/2 → y = cos(3π/2) = 0
- θ = 2π/3 → y = cos(2π) = 1

Plot these points and draw the wave.

✔ Answer for #2:
Amplitude = 1, Period = 2π/3

---

## ✔ Problem 3: `y = 2 sin(θ/2)`

➤ Amplitude:

A = 2 → Amplitude = |2| = 2

➤ Period:

B = 1/2 → Period = 2π / (1/2) = 4π

➤ Graphing Instructions:

- Sine wave starts at origin (0,0).
- One full cycle from θ = 0 to θ = 4π.
- Key points (divide period into 4 parts → each part = π):
- θ = 0 → y = 2·sin(0) = 0
- θ = π → y = 2·sin(π/2) = 2
- θ = 2π → y = 2·sin(π) = 0
- θ = 3π → y = 2·sin(3π/2) = -2
- θ = 4π → y = 2·sin(2π) = 0

Plot these and connect with a smooth sine curve.

✔ Answer for #3:
Amplitude = 2, Period = 4π

---

## ✔ Problem 4: `y = (1/2)·cos(2θ)`

➤ Amplitude:

A = 1/2 → Amplitude = |1/2| = 0.5

➤ Period:

B = 2 → Period = 2π / 2 = π

➤ Graphing Instructions:

- Cosine wave, starts at max: θ=0 → y=0.5
- One period from 0 to π.
- Key points (each quarter-period = π/4):
- θ = 0 → y = 0.5·cos(0) = 0.5
- θ = π/4 → y = 0.5·cos(π/2) = 0
- θ = π/2 → y = 0.5·cos(π) = -0.5
- θ = 3π/4 → y = 0.5·cos(3π/2) = 0
- θ = π → y = 0.5·cos(2π) = 0.5

Plot and connect.

✔ Answer for #4:
Amplitude = 0.5, Period = π

---

## 📊 Summary Table:

| Problem | Function | Amplitude | Period |
|---------|-------------------|-----------|--------|
| 1 | `y = 3 cos(2θ)` | 3 | π |
| 2 | `y = cos(3θ)` | 1 | 2π/3 |
| 3 | `y = 2 sin(θ/2)` | 2 | 4π |
| 4 | `y = (1/2) cos(2θ)` | 0.5 | π |

---

## 🖍️ How to Graph Each (Quick Tips):

- Use the x-axis labeled in radians as given on each grid.
- Mark the amplitude on the y-axis (positive and negative).
- Mark the period on the x-axis — that’s how long one full wave takes.
- Plot 5 key points per cycle: start, 1/4, 1/2, 3/4, end.
- Connect with smooth curves (no sharp corners!).

---

Let me know if you’d like me to sketch the graphs for you or provide printable answer keys! 😊