Trig Graphs - GCSE Maths - Steps, Examples & Worksheet - Free Printable

It looks like you've uploaded an educational image about trigonometric graphs, specifically showing the graphs of:

- $ y = \sin \theta $
- $ y = \cos \theta $
- $ y = \tan \theta $

This is a visual summary of key properties of trigonometric functions. Let's go through each graph and explain what it shows, then summarize the main concepts.

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🔹 1. Graph of $ y = \sin \theta $

Key Features:
- Period: $ 360^\circ $ (repeats every $ 360^\circ $)
- Amplitude: 1 (maximum value = 1, minimum value = -1)
- Shape: Smooth wave that starts at 0 when $ \theta = 0^\circ $
- Values:
- $ \sin(0^\circ) = 0 $
- $ \sin(90^\circ) = 1 $
- $ \sin(180^\circ) = 0 $
- $ \sin(270^\circ) = -1 $
- $ \sin(360^\circ) = 0 $

> ✔ The sine function is periodic with period $ 360^\circ $. It is continuous and smooth.

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🔹 2. Graph of $ y = \cos \theta $

Key Features:
- Period: $ 360^\circ $
- Amplitude: 1 (max = 1, min = -1)
- Shape: Smooth wave starting at 1 when $ \theta = 0^\circ $
- Values:
- $ \cos(0^\circ) = 1 $
- $ \cos(90^\circ) = 0 $
- $ \cos(180^\circ) = -1 $
- $ \cos(270^\circ) = 0 $
- $ \cos(360^\circ) = 1 $

> ✔ Like sine, cosine is periodic with period $ 360^\circ $, continuous, and oscillates between -1 and 1.

💡 Note: The cosine graph is a horizontal shift of the sine graph — specifically, $ \cos \theta = \sin(\theta + 90^\circ) $

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🔹 3. Graph of $ y = \tan \theta $

Key Features:
- Period: $ 180^\circ $ (repeats every $ 180^\circ $)
- No amplitude — the values go to infinity
- Vertical asymptotes occur where $ \cos \theta = 0 $, i.e., at:
- $ \theta = -90^\circ, 90^\circ, 270^\circ, \dots $
- Discontinuous at these asymptotes
- Behavior:
- As $ \theta $ approaches $ 90^\circ $ from the left, $ \tan \theta \to +\infty $
- As $ \theta $ approaches $ 90^\circ $ from the right, $ \tan \theta \to -\infty $
- Zero crossings: $ \tan \theta = 0 $ at $ \theta = 0^\circ, 180^\circ, 360^\circ, \dots $

> ⚠️ Unlike sine and cosine, tangent is not continuous due to vertical asymptotes every $ 180^\circ $.

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🧩 Summary of Key Concepts

| Function | Period | Range | Continuity | Asymptotes |
|--------|--------|-------|------------|------------|
| $ \sin \theta $ | $ 360^\circ $ | $[-1, 1]$ | Continuous | None |
| $ \cos \theta $ | $ 360^\circ $ | $[-1, 1]$ | Continuous | None |
| $ \tan \theta $ | $ 180^\circ $ | $(-\infty, \infty)$ | Not continuous | Every $ 90^\circ $ (odd multiples) |

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✔ Why This Matters

Understanding these graphs helps in:
- Solving trigonometric equations
- Graphing transformations (e.g., $ y = 2\sin(3\theta) $)
- Analyzing real-world periodic phenomena (sound waves, tides, etc.)

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📌 Final Takeaway

The image effectively teaches:
- Sine and cosine are smooth, continuous, periodic functions with period $ 360^\circ $, bounded between -1 and 1.
- Tangent has a shorter period ($ 180^\circ $), is unbounded, and has vertical asymptotes where cosine is zero.

Let me know if you'd like help solving a specific problem involving these graphs!