Welcome to Trigonometry!
Hello! Welcome to your study guide for Trigonometry. If the word "Trigonometry" makes you think of endless triangles and confusing buttons on your calculator, don't worry! At its heart, trigonometry is simply the study of how angles and sides of triangles relate to each other. It’s also about waves and patterns that repeat—which is why it’s used in everything from music production to designing rollercoasters.
In this guide, we will break down the AQA AS Level (Paper 2) requirements into bite-sized pieces. Whether you love math or find it a bit of a mountain to climb, we’ll get to the top together!
1. Beyond Right-Angled Triangles
You probably remember SOH CAH TOA from GCSE. While that's great for right-angled triangles, AS Level Trigonometry lets us work with any triangle. We have three main tools for this:
The Sine Rule
Use this when you have "matching pairs" (a side and its opposite angle).
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Example: If you know side \(a\) and angle \(A\), and you want to find side \(b\), you just need angle \(B\).
The Cosine Rule
Think of this as Pythagoras’ older brother. It works for triangles that don't have a right angle. Use it when you have two sides and the angle between them (SAS) or all three sides (SSS).
To find a side: \(a^2 = b^2 + c^2 - 2bc \cos A\)
To find an angle: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)
Area of a Triangle
Forget "half base times height" for a moment! If you know two sides and the included angle (the angle between them), you can find the area easily:
Area = \(\frac{1}{2} ab \sin C\)
Quick Review:
- Sine Rule: Needs pairs of sides/angles.
- Cosine Rule: Use for "Side-Angle-Side" or "Side-Side-Side".
- Calculator Check: Always make sure your calculator is in Degrees (D) mode unless the question specifically mentions radians!
Key Takeaway: These rules allow you to "solve" any triangle as long as you have at least three pieces of information.
2. Trigonometric Graphs and Symmetry
Trig functions aren't just for triangles; they are periodic, meaning they repeat their shape over and over like a wave.
The Big Three Graphs
1. Sine (\(y = \sin x\)): Starts at \((0,0)\), goes up to \(1\) at \(90^\circ\), and back to \(0\) at \(180^\circ\). It repeats every \(360^\circ\).
2. Cosine (\(y = \cos x\)): Looks just like the Sine wave but starts at its peak \((0,1)\). It also repeats every \(360^\circ\).
3. Tangent (\(y = \tan x\)): This one is different! It has asymptotes (lines the graph never touches) at \(90^\circ, 270^\circ\), etc. It repeats every \(180^\circ\).
Did you know?
The Cosine graph is just the Sine graph shifted to the left by \(90^\circ\). In math, we call this a phase shift!
Using Symmetry
Because these graphs repeat, an equation like \(\sin x = 0.5\) has many answers. Your calculator will only give you one (the principal value). To find the others, use the symmetry of the graph or the CAST diagram.
Common Mistake: Forgetting to look for a second solution! For \(\sin x\), the second solution is often \(180 - \theta\). For \(\cos x\), it’s often \(360 - \theta\).
Key Takeaway: Trig graphs are like wallpaper patterns; once you know one section, you can predict the rest of the wall!
3. Two Essential Identities
In AS Level, you need to memorize two "tricks" to simplify hard equations. These are called identities because they are true for every angle.
Identity 1: The Tangent Rule
\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
If you ever see \(\sin\) and \(\cos\) in the same equation, try dividing by \(\cos\) to turn it into \(\tan\).
Identity 2: The Squared Rule
\(\sin^2 \theta + \cos^2 \theta \equiv 1\)
This is actually just Pythagoras' Theorem hidden in a circle! You can rearrange it to:
\(\sin^2 \theta \equiv 1 - \cos^2 \theta\)
\(\cos^2 \theta \equiv 1 - \sin^2 \theta\)
Memory Aid: Think of \(\sin^2 \theta + \cos^2 \theta = 1\) as the "Home Base." Whenever an equation looks messy with squared terms, head back to home base to swap them out.
Key Takeaway: Identities are tools for "un-sticking" a problem. Use them to make an equation have only one type of trig function (e.g., all \(\sin\) or all \(\cos\)).
4. Solving Trigonometric Equations
Solving these is a step-by-step process. Don't worry if it seems tricky at first; it's all about the routine.
Step-by-Step Method:
1. Simplify: Use identities to get the equation into the form \(\sin x = k\), \(\cos x = k\), or \(\tan x = k\).
2. Calculator: Use the inverse function (e.g., \(\sin^{-1}\)) to find the first answer. This is your "base" angle.
3. Find Others: Use the graph symmetry or a CAST diagram to find all other angles within the interval given in the question (usually \(0^\circ \leq x \leq 360^\circ\)).
4. Check: Does each answer fit in the range?
Dealing with Multiple Angles (e.g., \(\sin 2\theta = 0.5\))
If the angle is \(2\theta\), you must change the interval first. If \(\theta\) is between \(0\) and \(360\), then \(2\theta\) is between \(0\) and \(720\). Find all solutions for \(2\theta\) first, then divide them all by 2 at the very end.
Quadratic Trig Equations
Sometimes you’ll see something like \(2\sin^2 x + \sin x - 1 = 0\).
Top Tip: Replace \(\sin x\) with \(y\). It becomes \(2y^2 + y - 1 = 0\). Solve this like a normal quadratic, then set your answers back to \(\sin x = \dots\)
Quick Review:
- Is it a quadratic? Substitute \(y\).
- Did I find all solutions in the range?
- Did I divide by the multiplier (like the \(2\) in \(2\theta\)) at the last step?
Key Takeaway: Solving trig equations is like a puzzle. Isolate the trig function, find the first piece, and use symmetry to find the rest.
Final Summary for Paper 2
To succeed in the trigonometry section of Paper 2, focus on these three pillars:
1. Triangles: Knowing when to use Sine vs. Cosine rules.
2. Identities: Swapping \(\tan\) for \(\sin/\cos\) and using \(\sin^2 + \cos^2 = 1\).
3. Equations: Being methodical with your calculator and finding multiple solutions within the range.
You've got this! Practice a few triangle problems and sketching the graphs, and you'll be a trig expert in no time.