Graphs of the basic trigonometric functions

Introduction to Trigonometric Graphs

Welcome! In this chapter, we are going to explore the "blueprints" of trigonometry: the graphs of sine, cosine, and tangent. These aren't just squiggly lines on a page; they represent patterns that repeat forever, just like the tides of the ocean or the vibration of a guitar string.

Don't worry if you find graphs a bit intimidating at first. We will break them down into simple shapes and show you the "magic numbers" that help you draw them perfectly every time.

1. The Sine Graph: \(y = \sin \theta\)

The sine graph is often called a sine wave. It is a smooth, continuous curve that flows up and down.

Key Features to Remember:

• Starting Point: It starts at the origin \((0, 0)\).
• The Peaks: The highest it ever goes is 1 (at \(90^\circ\)).
• The Valleys: The lowest it ever goes is -1 (at \(270^\circ\)).
• The Crossing Points: It crosses the horizontal axis at \(0^\circ, 180^\circ,\) and \(360^\circ\).
• Periodicity: The pattern repeats every \(360^\circ\). This length is called the period.

Symmetry Trick:

The sine graph is very symmetrical. For example, \(\sin(30^\circ)\) is the same as \(\sin(150^\circ)\). You can find this by doing \(180^\circ - \theta\). If you know the value in the first \(90^\circ\), you can find the others using the shape of the wave!

Quick Review: The sine graph looks like a snake starting at 0, going up to 1, back to 0, down to -1, and back to 0.

2. The Cosine Graph: \(y = \cos \theta\)

The cosine graph looks almost exactly like the sine graph, but it's "shifted." If sine is a snake starting at the ground, cosine is a snake starting at the top of a fence.

Key Features to Remember:

• Starting Point: It starts at its maximum value: \((0, 1)\).
• The Shape: It looks like a "valley" or a "bucket" between \(0^\circ\) and \(360^\circ\).
• The Crossing Points: It crosses the horizontal axis at \(90^\circ\) and \(270^\circ\).
• The Bottom: It hits its lowest point, -1, at \(180^\circ\).
• Periodicity: Just like sine, its period is \(360^\circ\).

Comparison Analogy:

Imagine the sine and cosine graphs are two siblings running a race. Cosine had a \(90^\circ\) head start! In fact, \(\cos \theta = \sin(\theta + 90^\circ)\).

Quick Review: Cosine starts at 1, drops to 0 at \(90^\circ\), hits -1 at \(180^\circ\), and climbs back up to 1 at \(360^\circ\).

3. The Tangent Graph: \(y = \tan \theta\)

The tangent graph is the "rebel" of the group. It doesn't look like a wave at all. Instead, it consists of several separate branches.

Key Features to Remember:

• Asymptotes: These are vertical lines that the graph gets closer and closer to but never touches. For \(\tan \theta\), these happen at \(90^\circ\) and \(270^\circ\). If you type \(\tan(90)\) into your calculator, it will give you an error!
• Periodicity: Unlike sine and cosine, the tangent graph repeats every \(180^\circ\).
• Range: While sine and cosine are trapped between -1 and 1, tangent goes off to positive infinity and negative infinity.

Memory Aid:

Think of the tangent graph as a series of "S" shapes separated by electric fences (the asymptotes). The "S" always passes through \(0\) exactly halfway between the fences.

Key Takeaway: Tangent is different! It repeats twice as often (every \(180^\circ\)) and has "break points" where the function doesn't exist.

4. Exact Values You Must Know

The OCR syllabus requires you to know the exact values for certain angles without using a calculator. These are your "bread and butter" for exam questions.

Common Sine and Cosine Values:

• \(\sin(0^\circ) = 0\) | \(\cos(0^\circ) = 1\)
• \(\sin(30^\circ) = \frac{1}{2}\) | \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
• \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\) | \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) | \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(90^\circ) = 1\) | \(\cos(90^\circ) = 0\)

Common Tangent Values:

• \(\tan(0^\circ) = 0\)
• \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
• \(\tan(45^\circ) = 1\)
• \(\tan(60^\circ) = \sqrt{3}\)

Did you know? You can remember the sine values for \(0, 30, 45, 60, 90\) by looking at the numerators: \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\). It’s a perfect sequence!

5. Using Symmetries to Solve Problems

Because these graphs repeat, there are often two angles that give the same result. For example, if \(\sin \theta = 0.5\), \(\theta\) could be \(30^\circ\) or \(150^\circ\).

Step-by-Step: Finding other angles

1. Find the principal value (the one your calculator gives you).
2. For Sine: The second value is \(180^\circ - \text{principal value}\).
3. For Cosine: The second value is \(360^\circ - \text{principal value}\). (You can also use \(-\theta\) because the graph is symmetrical across the y-axis).
4. For Tangent: Just add or subtract \(180^\circ\) to find more values.

Common Mistakes to Avoid

• Calculator Mode: Always check if your calculator is in Degrees (D). If you see an 'R' (Radians), your answers will be wrong for this chapter!
• Asymptotes: Don't let your tangent graph touch the lines at \(90^\circ\) or \(270^\circ\). It should curve towards them but stay forever separate.
• Sine vs Cosine: Remember that Sine starts at 0 and Cosine starts at 1. A common mistake is to swap their starting points.

Summary: Mastery of these graphs comes from practice. Try sketching all three from memory for the range \(-360^\circ \leq \theta \leq 360^\circ\). Once you can see the wave in your head, the equations become much easier!