sin, cos and tan for all arguments

Introduction: Beyond the Right-Angled Triangle

Welcome! So far in your mathematical journey, you have likely used trigonometry to find missing sides and angles in right-angled triangles. You probably remember SOH CAH TOA. But what happens if the angle is bigger than \(90^\circ\)? What if it is negative?

In this chapter, we are going to unlock "Trigonometry for all arguments." This means we will learn how sin, cos, and tan behave for any angle you can imagine. This is vital for understanding everything from the way sound waves travel to how planets orbit the sun!

1. The Unit Circle: Our Mathematical Map

The biggest hurdle in moving beyond GCSE is realizing that sin, cos, and tan aren't just ratios in a triangle; they are coordinates on a circle.

Imagine a circle with a radius of \(1\) unit, centered at the origin \((0,0)\). We call this the Unit Circle.

If we pick a point \(P\) on the edge of this circle at an angle \(\theta\) (measured anticlockwise from the positive x-axis):

1. The x-coordinate of the point is \(\cos \theta\)
2. The y-coordinate of the point is \(\sin \theta\)
3. The gradient (steepness) of the line from the center to the point is \(\tan \theta\)

Quick Review: Because the radius is \(1\), the maximum value for sin and cos is \(1\), and the minimum is \(-1\). They just keep looping around the circle!

2. The "CAST" Diagram (Signs of Trig Functions)

As you move around the circle, your \(x\) and \(y\) coordinates change from positive to negative. This tells us whether sin, cos, or tan will be positive or negative in the four quadrants of the circle.

We use the mnemonic CAST (starting from the bottom-right and going anticlockwise) or ASTC (starting top-right):

Quadrant 1 (\(0^\circ\) to \(90^\circ\)): All are positive.
Quadrant 2 (\(90^\circ\) to \(180^\circ\)): Only Sine is positive.
Quadrant 3 (\(180^\circ\) to \(270^\circ\)): Only Tangent is positive.
Quadrant 4 (\(270^\circ\) to \(360^\circ\)): Only Cosine is positive.

Memory Aid: Try the phrase "Add Sugar To Coffee" to remember the order (Quadrants 1, 2, 3, 4).

Example: If you calculate \(\sin 150^\circ\), the answer will be positive because \(150^\circ\) is in the 2nd Quadrant. However, \(\cos 150^\circ\) will be negative.

3. Exact Values You Need to Know

The OCR syllabus requires you to know the exact values for specific angles. Using a calculator is fine, but you must recognize these "surd" forms instantly.

The "Big Five" Angles:

- \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 60^\circ = \frac{1}{2}\)
- \(\sin 45^\circ = \frac{\sqrt{2}}{2}\) and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\tan 30^\circ = \frac{\sqrt{3}}{3}\), \(\tan 45^\circ = 1\), \(\tan 60^\circ = \sqrt{3}\)

Did you know? There is a pattern! \(\sin 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) is just \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\). Simplify them, and you have your values!

4. Trigonometric Graphs

Visualizing the graphs is the best way to understand periodicity (how often the graph repeats) and symmetry.

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

- Starts at \((0,0)\).
- Looks like a smooth wave.
- Period: \(360^\circ\) (it repeats every \(360^\circ\)).
- Symmetry: \(\sin \theta = \sin(180 - \theta)\).

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

- Starts at \((0,1)\).
- It is just the sine graph shifted \(90^\circ\) to the left!
- Period: \(360^\circ\).
- Symmetry: \(\cos \theta = \cos(-\theta)\) or \(\cos \theta = \cos(360 - \theta)\).

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

- Looks very different—it has asymptotes (lines the graph never touches) at \(90^\circ, 270^\circ\), etc.
- Period: \(180^\circ\) (it repeats twice as often as sin and cos!).

Key Takeaway: Because these graphs repeat, an equation like \(\sin \theta = 0.5\) has infinitely many solutions. We usually only look for the ones within a specific range, like \(0 \le \theta < 360\).

5. Symmetry and Negative Arguments

Don't worry if you see an angle like \(-30^\circ\). A negative angle just means you are measuring clockwise instead of anticlockwise on your circle.

Simple Rules to Remember:

- \(\sin(-\theta) = -\sin \theta\)
- \(\cos(-\theta) = \cos \theta\) (Cosine "swallows" the negative sign!)
- \(\tan(-\theta) = -\tan \theta\)

Example: \(\cos(-60^\circ)\) is exactly the same as \(\cos(60^\circ)\), which is \(0.5\).

Common Mistakes to Avoid

- Calculator Mode: Always check if your calculator is in Degrees (D) or Radians (R). For this section of the AS course, you will mostly use Degrees.
- Missing Solutions: When solving \(\sin \theta = 0.5\), your calculator gives \(30^\circ\). Don't forget the second solution: \(180 - 30 = 150^\circ\)! Always look at the graph or CAST diagram to find the "hidden" second angle.
- Tan Asymptotes: Remember that \(\tan 90^\circ\) is undefined. If you type it into your calculator, you will get a "Math Error." This is normal!

Final Quick Review

- Sine is the height (\(y\)) on the unit circle.
- Cosine is the width (\(x\)) on the unit circle.
- Tan is the gradient (\(y/x\)).
- Use CAST to find the sign of the result.
- Use graph symmetry to find all possible angles in a range.