worksheet graphs of trig functions Doc Template | pdfFiller - Free Printable

You’re working on a trigonometry worksheet focused on graphing the basic sine and cosine functions over one full period: from –2π to 2π.

Let’s solve both problems step by step, filling in the tables, then identifying domain, range, and period.

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## 🔹 PROBLEM 1: f(x) = sin x

📊 Step 1: Complete the table for f(x) = sin x

We’ll evaluate sin x at key angles between –2π and 2π. These are standard unit circle values.

| x | f(x) = sin x |
|----------|--------------|
| –2π | 0 |
| –3π/2 | 1 |
| –π | 0 |
| –π/2 | –1 |
| 0 | 0 |
| π/2 | 1 |
| π | 0 |
| 3π/2 | –1 |
| 2π | 0 |

✔ Explanation:
- Sine is periodic with period 2π, so it repeats every 2π.
- At x = 0, sin(0) = 0.
- At x = π/2, sin(π/2) = 1 (maximum).
- At x = π, sin(π) = 0.
- At x = 3π/2, sin(3π/2) = –1 (minimum).
- At x = 2π, sin(2π) = 0 → back to start.
- Same pattern repeats backward: sin(–π/2) = –1, sin(–π) = 0, etc.

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📈 Step 2: Graph f(x) = sin x

You should draw a smooth wave starting at (–2π, 0), going up to (–3π/2, 1), down to (–π, 0), down to (–π/2, –1), up to (0, 0), up to (π/2, 1), down to (π, 0), down to (3π/2, –1), and back to (2π, 0).

It’s a classic sine wave oscillating between –1 and 1.

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✔ Step 3: Domain, Range, Period

- Domain: All real numbers → (–∞, ∞)
*(Sine is defined for all x)*

- Range: Values between –1 and 1 → [–1, 1]
*(Sine never exceeds these bounds)*

- Period: Distance for one full cycle → 2π
*(Repeats every 2π units)*

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## 🔹 PROBLEM 2: f(x) = cos x

📊 Step 1: Complete the table for f(x) = cos x

Cosine also has period 2π. Let’s compute values:

| x | f(x) = cos x |
|----------|--------------|
| –2π | 1 |
| –3π/2 | 0 |
| –π | –1 |
| –π/2 | 0 |
| 0 | 1 |
| π/2 | 0 |
| π | –1 |
| 3π/2 | 0 |
| 2π | 1 |

✔ Explanation:
- Cosine starts at 1 when x = 0.
- At x = π/2 → 0
- At x = π → –1 (minimum)
- At x = 3π/2 → 0
- At x = 2π → 1 (back to start)
- Symmetric: cos(–x) = cos(x), so values at negative angles mirror positive ones.

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📈 Step 2: Graph f(x) = cos x

Start at (–2π, 1), go down to (–3π/2, 0), down to (–π, –1), up to (–π/2, 0), up to (0, 1), down to (π/2, 0), down to (π, –1), up to (3π/2, 0), and back to (2π, 1).

This is a cosine wave, which looks like a sine wave shifted left by π/2.

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✔ Step 3: Domain, Range, Period

- Domain: All real numbers → (–∞, ∞)
*(Cosine is defined everywhere)*

- Range: Values between –1 and 1 → [–1, 1]
*(Same as sine — bounded between –1 and 1)*

- Period: Distance for one full cycle → 2π
*(Repeats every 2π units)*

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## 🧾 FINAL ANSWERS TO FILL IN WORKSHEET

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1. f(x) = sin x

Table:

| x | f(x) |
|----------|------|
| –2π | 0 |
| –3π/2 | 1 |
| –π | 0 |
| –π/2 | –1 |
| 0 | 0 |
| π/2 | 1 |
| π | 0 |
| 3π/2 | –1 |
| 2π | 0 |

Domain: (–∞, ∞)
Range: [–1, 1]
Period: 2π

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2. f(x) = cos x

Table:

| x | f(x) |
|----------|------|
| –2π | 1 |
| –3π/2 | 0 |
| –π | –1 |
| –π/2 | 0 |
| 0 | 1 |
| π/2 | 0 |
| π | –1 |
| 3π/2 | 0 |
| 2π | 1 |

Domain: (–∞, ∞)
Range: [–1, 1]
Period: 2π

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## 💡 BONUS TIP: Memorization

> Sine starts at 0, goes up to max, back to 0, down to min, back to 0.
> Cosine starts at max, goes down to 0, to min, back to 0, back to max.

Both have:
- Amplitude = 1
- Period = 2π
- No phase shift or vertical shift in these basic forms.

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✔ You’re now ready to graph both functions accurately on the grid provided! Just plot the points from the tables and connect them smoothly with a wave-like curve.

Let me know if you want help sketching the graphs or understanding transformations later!