Welcome to Trigonometric Functions!
In your earlier years of school, you probably learned about trigonometry using right-angled triangles and the famous SOH CAH TOA. While that's a great start, the world isn't made only of triangles! Trigonometry is actually the study of periodic functions—things that repeat in cycles, like sound waves, tides, or even your heartbeat.
In this chapter, we are going to move beyond triangles and look at how Sine, Cosine, and Tangent work for any angle you can imagine. Don't worry if it seems like a big jump; we'll take it one step at a time!
1. Extending Trigonometry: The Unit Circle
How can we have an angle bigger than 90° in a triangle? We can’t! To solve this, mathematicians use the Unit Circle. This is just a circle with a radius of 1, centered at the origin (0,0) on a graph.
How it works:
Imagine a point moving around the edge of this circle. The angle \( \theta \) (theta) starts from the positive x-axis and moves anti-clockwise.
The coordinates of this point are very special:
- The x-coordinate is always \( \cos \theta \).
- The y-coordinate is always \( \sin \theta \).
- The gradient (slope) of the line from the center to the point is \( \tan \theta \).
Quick Review:
- \( \sin \theta = y \)
- \( \cos \theta = x \)
- \( \tan \theta = \frac{y}{x} \)
The CAST Diagram (Finding Angles in different quadrants)
Since the circle has four quadrants, the "signs" (positive or negative) of sin, cos, and tan change. A common memory aid is CAST (starting from the bottom-right and moving anti-clockwise):
- C (4th Quadrant, 270°-360°): Only Cos is positive.
- A (1st Quadrant, 0°-90°): All (Sin, Cos, Tan) are positive.
- S (2nd Quadrant, 90°-180°): Only Sin is positive.
- T (3rd Quadrant, 180°-270°): Only Tan is positive.
Analogy: Think of the CAST diagram like a compass for signs. It tells you exactly where sin, cos, and tan "feel at home" (positive) and where they feel "out of place" (negative).
Key Takeaway: The Unit Circle allows us to define trig functions for any angle, even negative ones or those much larger than 360°.
2. Exact Values You Need to Know
In your exams, you'll often be asked for "exact values." This means no decimals! You need to memorize these for 0°, 30°, 45°, 60°, and 90°.
The Cheat Sheet Table:
- \( \sin(30^\circ) = \frac{1}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
- \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \), \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \), \( \tan(45^\circ) = 1 \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), \( \cos(60^\circ) = \frac{1}{2} \), \( \tan(60^\circ) = \sqrt{3} \)
Did you know? There is a finger trick for this! If you hold up your left hand (palm facing you) and fold down a finger, you can count the fingers on either side to find sin and cos. Ask your teacher to show you the "Trig Hand Trick" – it's a lifesaver!
Key Takeaway: Always check if the question asks for an "exact value." If it does, keep those square roots in your answer!
3. Trigonometric Graphs
If we plot the values of sin, cos, and tan as we move around the circle, we get wave-like graphs. You should be able to sketch these by hand.
The Sine Wave \( y = \sin \theta \)
- Starts at (0,0).
- Reaches a maximum of 1 at 90°.
- Crosses the x-axis at 0°, 180°, and 360°.
- Period: 360° (the wave repeats every 360°).
- Amplitude: 1 (it goes 1 unit up and 1 unit down from the center).
The Cosine Wave \( y = \cos \theta \)
- Starts at its maximum (0,1).
- It looks exactly like a sine wave, just shifted 90° to the left!
- Period: 360°.
- Amplitude: 1.
The Tangent Graph \( y = \tan \theta \)
- This one is the "rebel." It doesn't look like a wave; it looks like a series of curves.
- It has asymptotes at 90° and 270°. These are vertical lines that the graph gets closer and closer to but never touches.
- Period: 180° (it repeats twice as fast as sin and cos!).
Transformations:
You might be asked to sketch variations. For example:
- \( y = 3\sin \theta \) is a stretch vertically (makes the wave taller).
- \( y = \cos(\theta + 30^\circ) \) is a translation (shifts the wave 30° to the left).
- \( y = -\sin \theta \) is a reflection in the x-axis (flips it upside down).
Key Takeaway: Learn the "shape" and key points (0, 90, 180, 270, 360) of the three basic graphs. Everything else is just a tweak of these shapes.
4. Trigonometric Identities
An identity is an equation that is true for every value of \( \theta \). You need to know two main ones for your AS Level:
Identity 1: The Tangent Identity
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
This is useful when you have an equation with sin, cos, and tan mixed together. You can replace tan to simplify things.
Identity 2: The Pythagorean Identity
\( \sin^2 \theta + \cos^2 \theta = 1 \)
Common Mistake: Be careful with notation! \( \sin^2 \theta \) means \( (\sin \theta)^2 \). It does not mean \( \sin(\theta^2) \).
Analogy: Think of these identities like "currency exchange." If you have sin but you need cos, these formulas let you swap one for the other.
Key Takeaway: If you see a \( \sin^2 \theta \) in a problem, your brain should immediately think "maybe I can use \( 1 - \cos^2 \theta \)."
5. Solving Trigonometric Equations
This is where everything comes together. You'll be asked to find values of \( \theta \) that make an equation true within a specific range (usually 0° to 360°).
The Step-by-Step Process:
1. Isolate the trig function: Get it into the form \( \sin \theta = \text{number} \).
2. Find the Principal Value (PV): Use your calculator (e.g., \( \theta = \sin^{-1}(0.5) \)).
3. Find the other values: Use your CAST diagram or the symmetry of the graph.
4. Check the range: Make sure your answers are within the interval asked for (e.g., \( 0 \le \theta \le 360 \)).
Solving Quadratic Trig Equations:
Sometimes you'll see something like \( 2\cos^2 \theta - \cos \theta - 1 = 0 \).
Don't panic! This is just a quadratic equation in disguise.
- Let \( x = \cos \theta \).
- Rewrite it as \( 2x^2 - x - 1 = 0 \).
- Factorise it like a normal quadratic: \( (2x + 1)(x - 1) = 0 \).
- Then solve: \( \cos \theta = -0.5 \) and \( \cos \theta = 1 \).
Equations with Multiples:
If you have \( \sin 2\theta = 0.5 \), solve for \( 2\theta \) first, and remember to expand your range. If \( \theta \) is between 0 and 360, then \( 2\theta \) must be between 0 and 720. Once you find all the values for \( 2\theta \), divide them all by 2 at the very end.
Key Takeaway: Most trig equations have more than one answer. Always check your graph or CAST diagram to find that hidden second (or third) solution!
Summary Checklist
Before you move on, make sure you can:
- [ ] Explain sin, cos, and tan using the Unit Circle.
- [ ] Sketch the graphs of \( \sin \theta, \cos \theta, \) and \( \tan \theta \).
- [ ] State the exact values for 30°, 45°, and 60°.
- [ ] Use the two main identities to simplify expressions.
- [ ] Solve trig equations and find all solutions in a given interval.