Welcome to P2 Trigonometry!
In Pure Mathematics 1 (P1), you learned the basics of triangles, the sine and cosine rules, and how to draw the curvy waves of the trig graphs. Now that we are in Unit P2, we are going to level up!
In this chapter, we will learn how to simplify messy equations using Trigonometric Identities and how to solve equations to find missing angles in both degrees and radians. Don't worry if trigonometry felt like a maze before; we’re going to break it down into simple, logical steps.
Why does this matter? Trigonometry isn't just for triangles in textbooks. It’s used to map out the stars, design video game graphics, and even help your GPS figure out exactly where you are on Earth!
1. The Golden Rules: Trigonometric Identities
An "identity" in math is like a rule that is always true, no matter what number you plug in. In P2, there are two main identities you must know by heart. They are your best friends for simplifying equations.
Identity #1: The Tangent Rule
\( \tan \theta \equiv \frac{\sin \theta}{\cos \theta} \)
This means if you know the sine and cosine of an angle, you automatically know the tangent!
Example: If \( \sin \theta = 0.6 \) and \( \cos \theta = 0.8 \), then \( \tan \theta = \frac{0.6}{0.8} = 0.75 \).
Identity #2: The Pythagorean Identity
\( \sin^2 \theta + \cos^2 \theta \equiv 1 \)
This comes from Pythagoras' Theorem. It is incredibly useful because it allows you to swap sines for cosines.
You can rearrange it to get:
1. \( \sin^2 \theta = 1 - \cos^2 \theta \)
2. \( \cos^2 \theta = 1 - \sin^2 \theta \)
Quick Tip:
In math, \( \sin^2 \theta \) is just a shorter way of writing \( (\sin \theta)^2 \). It means you find the sine of the angle first, then square the result.
Key Takeaway: Identities are tools used to turn an equation with two different trig functions (like \( \sin \) and \( \cos \)) into an equation with just one.
2. Solving Simple Trig Equations
Solving an equation like \( \sin x = 0.5 \) is basically asking: "At what angle does the height of the wave reach 0.5?"
The Step-By-Step Process
Step 1: Isolate the trig function. Get \( \sin x \), \( \cos x \), or \( \tan x \) by itself on one side of the equals sign.
Step 2: Find the Principal Value (PV). Use your calculator to find the first angle. For \( \sin x = 0.5 \), you would type \( \sin^{-1}(0.5) \) to get \( 30^\circ \).
Step 3: Find the other values. Trigonometric waves repeat! There is usually more than one answer in a given range (like \( 0^\circ \) to \( 360^\circ \)). You can find these using the CAST diagram or the Trig Graphs.
Did you know?
The CAST diagram (sometimes called the Unit Circle) helps you remember which functions are positive in which quadrant:
Cosine is positive in the 4th quadrant (\( 270^\circ \) to \( 360^\circ \))
All are positive in the 1st quadrant (\( 0^\circ \) to \( 90^\circ \))
Sine is positive in the 2nd quadrant (\( 90^\circ \) to \( 180^\circ \))
Tangent is positive in the 3rd quadrant (\( 180^\circ \) to \( 270^\circ \))
Mnemonic: All Silver Tea Cups or CAST.
Quick Review:
For \( \sin x = k \): Second value is \( 180^\circ - PV \)
For \( \cos x = k \): Second value is \( 360^\circ - PV \)
For \( \tan x = k \): Second value is \( 180^\circ + PV \)
3. Dealing with Substituted Arguments
Sometimes the exam will give you something like \( \cos(x + 30^\circ) = 0.5 \) or \( \tan(2x) = 1 \). Don't panic! Treat the part inside the bracket as a single "blob."
How to solve \( \tan(2x) = 1 \) for \( 0^\circ \leq x \leq 180^\circ \):
1. Change the Range: If the range for \( x \) is \( 0 \) to \( 180 \), then the range for \( 2x \) is \( 0 \) to \( 360 \).
2. Solve for the "blob": \( 2x = \tan^{-1}(1) = 45^\circ \).
Using the \( \tan \) rule (add \( 180 \)), the next value for \( 2x \) is \( 45 + 180 = 225^\circ \).
3. Solve for \( x \): Divide your answers by 2.
\( x = 22.5^\circ \) and \( x = 112.5^\circ \).
Common Mistake: Forgetting to change the range. If you don't look far enough, you'll miss half of your answers!
4. Quadratic Trig Equations
Sometimes trig equations look like quadratic equations, such as:
\( 6\cos^2 x + \sin x - 5 = 0 \)
To solve these, we need one type of trig function. Notice we have a \( \cos^2 x \) and a \( \sin x \). We can change the \( \cos^2 x \) using our identity!
Step-by-Step Example:
1. Substitute: Replace \( \cos^2 x \) with \( (1 - \sin^2 x) \).
\( 6(1 - \sin^2 x) + \sin x - 5 = 0 \)
2. Expand and Simplify:
\( 6 - 6\sin^2 x + \sin x - 5 = 0 \)
\( -6\sin^2 x + \sin x + 1 = 0 \)
Multiply by -1 to make it look nicer: \( 6\sin^2 x - \sin x - 1 = 0 \)
3. Factorise: This is just like \( 6y^2 - y - 1 = 0 \).
\( (3\sin x + 1)(2\sin x - 1) = 0 \)
4. Solve:
Either \( \sin x = -1/3 \) or \( \sin x = 1/2 \).
Now solve both to find all possible values of \( x \) in your range.
Key Takeaway: If you see a "squared" trig term and a "normal" one, use \( \sin^2 \theta + \cos^2 \theta = 1 \) to make them match before solving.
5. Working with Radians
In P2, questions often ask for answers in radians rather than degrees.
Analogy: Degrees and Radians are like Celsius and Fahrenheit. They measure the same thing, just on a different scale!
Important reminders for Radians:
- Switch your calculator: Make sure there is a little "R" on the screen, not a "D".
- The Magic Number: \( 180^\circ \) is the same as \( \pi \) radians.
- The Full Circle: \( 360^\circ \) is the same as \( 2\pi \) radians.
If you are solving in radians, the rules for finding "other values" change slightly:
For \( \sin x \): Use \( \pi - PV \)
For \( \cos x \): Use \( 2\pi - PV \)
For \( \tan x \): Use \( \pi + PV \)
Summary Checklist
1. Do I know \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sin^2 \theta + \cos^2 \theta = 1 \)?
2. Is my calculator in the right mode (Degrees or Radians)?
3. Have I found all the values in the range, not just the principal one?
4. If the argument is modified (like \( 2x \)), did I adjust my range before solving?
Don't worry if this seems tricky at first! Trigonometry is a skill that gets much easier with practice. Start with the simple equations, and soon the complex ones will feel like second nature.