sin, cos and tan for all arguments

Welcome to the World of Circular Trigonometry!

Up until now, you might have thought of sine, cosine, and tangent as things that only live inside right-angled triangles. But what happens if you have an angle of \(120^\circ\), or even a negative angle like \(-45^\circ\)? You can't put those in a right-angled triangle!

In this chapter, we are going to "unlock" trigonometry so it works for any angle (or "argument") you can imagine. We do this by moving away from triangles and moving onto circles. Don't worry if this seems a bit strange at first—once you see the pattern, it’s like learning the secret code to the universe!

1. The Unit Circle: Trigonometry's Home

To understand trig for all angles, we use the Unit Circle. This is simply a circle with a radius of \(1\), centered at the origin \((0,0)\) on a graph.

Imagine a point \(P\) moving around this circle. The angle \(\theta\) always starts from the positive x-axis and turns anti-clockwise.

• The x-coordinate of the point is always \(\cos \theta\).
• The y-coordinate of the point is always \(\sin \theta\).
• The tangent is the gradient of the line from the center to the point: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Analogy: Think of the unit circle like a clock. Instead of telling time, the "hands" tell us the value of sine and cosine based on where they are pointing!

Quick Review Box:
Positive angles: Turn anti-clockwise.
Negative angles: Turn clockwise.
The Radius: Always \(1\).

Key Takeaway: \(\cos \theta\) is how far across you are; \(\sin \theta\) is how far up or down you are.

2. The Four Quadrants (The CAST Diagram)

Because the point is moving around a graph, the values of \(\sin\), \(\cos\), and \(\tan\) will be positive or negative depending on which "neighborhood" (quadrant) the angle is in.

The Quadrants:

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

Memory Aid: Use the mnemonic CAST (starting from the bottom-right and going anti-clockwise) or All Silver Tea Cups (starting from the top-right and going anti-clockwise).

Did you know? This explains why \(\sin(150^\circ)\) is positive (it's in the Sine quadrant), but \(\cos(150^\circ)\) is negative!

Key Takeaway: The quadrant tells you whether your answer should be plus or minus. Always check this first!

3. Finding Values for Any Angle

How do we calculate \(\cos(210^\circ)\) without just stabbing buttons on a calculator? We use the Reference Angle (or principal angle).

Step-by-Step Process:

1. Sketch the angle: See which quadrant it lands in.
2. Find the acute angle to the x-axis: This is your "reference angle." (Always go to the horizontal x-axis, never the vertical y-axis!)
3. Apply the CAST rule: Decide if the result is \(+\) or \(-\).
4. Combine: Write your final answer using the trig value of the reference angle.

Example: Find \(\tan(300^\circ)\).
1. \(300^\circ\) is in the 4th quadrant.
2. The angle to the x-axis (\(360^\circ\)) is \(360 - 300 = 60^\circ\).
3. In the 4th quadrant, only Cosine is positive, so Tangent is negative.
4. Answer: \(-\tan(60^\circ) = -\sqrt{3}\).

Common Mistake to Avoid: Students often find the angle to the y-axis by mistake. Always measure your reference angle from the horizontal x-axis!

4. Exact Values You Must Know

The OCR syllabus requires you to know the exact values for certain angles. These are your "bread and butter" for exam questions.

The "Square Root Trick" to remember them:
Write down \(0, 1, 2, 3, 4\).
Square root them all: \(\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}\).
Divide by \(2\): \(\frac{0}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{2}{2}\).
This gives you the Sine values for \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) !

Summary Table (Degrees):

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

Stage 2 Extension: For A Level, you must also know these in Radians (\(\pi = 180^\circ\)).
Example: \(30^\circ = \frac{\pi}{6}\), \(45^\circ = \frac{\pi}{4}\), \(60^\circ = \frac{\pi}{3}\).

5. Graphs, Symmetry, and Periodicity

Trig functions repeat themselves. This is called periodicity.

• Sine and Cosine: Repeat every \(360^\circ\) (or \(2\pi\)).
• Tangent: Repeats every \(180^\circ\) (or \(\pi\)).

Symmetry Tricks:

The graphs of \(\sin\) and \(\cos\) are very curvy and symmetrical, which leads to these helpful rules:
1. Sine is Odd: \(\sin(-\theta) = -\sin(\theta)\)
2. Cosine is Even: \(\cos(-\theta) = \cos(\theta)\) (The minus sign just disappears!)
3. Supplementary Sine: \(\sin(180^\circ - \theta) = \sin \theta\)

Encouragement: Graphs are your best friend! If you get stuck on a calculation, a quick sketch of the wave will often show you where the positive and negative values are.

Key Takeaway: Because the graphs repeat forever, there are usually infinitely many angles that give the same trig value. In exams, you will usually be given a specific range, like \(0 \le \theta < 360^\circ\).

Final Checklist for Success:

1. Can you draw the Unit Circle and label \(\cos\) and \(\sin\)?
2. Do you know your CAST diagram by heart?
3. Can you find the reference angle to the x-axis?
4. Have you memorized the exact values for \(30^\circ, 45^\circ, 60^\circ\)?

If you can do those four things, you've mastered the foundations of trigonometry for all arguments!