Welcome to Trigonometry!
Hello! Welcome to one of the most exciting and useful chapters in AS Mathematics. If you’ve ever wondered how GPS systems pinpoint your location, how architects design curved buildings, or how music producers visualize sound waves, the answer is Trigonometry.
Don’t worry if you found triangles a bit confusing in the past. We are going to break this down step-by-step, starting with the basics you know and moving into the "wavey" world of functions and identities. Let’s dive in!
1. Beyond the Right-Angled Triangle
At GCSE, you learned SOH CAH TOA for right-angled triangles. At AS Level, we expand this. We look at "all arguments," which is just a fancy way of saying we can find the sine, cosine, or tangent of any angle—even negative ones or angles much larger than \( 180^\circ \).
The Sine and Cosine Rules
When we don't have a right angle, we use these two powerful tools:
The Sine Rule: Use this when you have "matching pairs" (an angle and its opposite side).
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
The Cosine Rule: Use this when you have "SAS" (Side-Angle-Side) or "SSS" (all three sides).
\( a^2 = b^2 + c^2 - 2bc \cos A \)
Area of a Triangle
You no longer need the vertical height to find the area! If you have two sides and the angle between them, use:
Area = \( \frac{1}{2} ab \sin C \)
Memory Aid: Think of the Sine Rule as the "Partner Rule"—you need to know a side and its partner angle to make it work!
Quick Takeaway: If the triangle has a "pair" of a side and an opposite angle, use the Sine Rule. If it looks like a "side-angle-side" sandwich, use the Cosine Rule.
2. Trigonometric Graphs and Symmetry
Trigonometry isn't just about static triangles; it’s about periodic functions—patterns that repeat forever.
The Big Three Graphs
1. The Sine Graph \( y = \sin x \): Starts at \( (0,0) \), goes up to \( 1 \), down to \( -1 \), and returns to \( 0 \) at \( 360^\circ \). It looks like a smooth "S" on its side.
2. The Cosine Graph \( y = \cos x \): Starts at its peak \( (0,1) \), drops to \( -1 \), and climbs back to \( 1 \) at \( 360^\circ \). It looks like a "Bucket" or a "Valley".
3. The Tangent Graph \( y = \tan x \): This one is the "rebel." It has asymptotes (invisible lines it never touches) at \( 90^\circ \) and \( 270^\circ \). It repeats every \( 180^\circ \).
Did you know? The period of a function is how often it repeats. For \( \sin \) and \( \cos \), the period is \( 360^\circ \). For \( \tan \), the period is \( 180^\circ \).
Symmetry and CAST
Because these graphs repeat, an equation like \( \sin x = 0.5 \) has more than one answer! You can find the first answer using your calculator, and the second answer using the graph's symmetry:
For Sine: \( 180 - \theta \)
For Cosine: \( 360 - \theta \)
For Tangent: \( 180 + \theta \)
Quick Takeaway: Always check your "interval" (e.g., \( 0 \le x \le 360 \)). Your calculator only gives you the Principal Value—you must find the others yourself!
3. Trigonometric Identities
In math, an identity is a statement that is true for every value of \( \theta \). Think of them as nicknames; "Sine over Cosine" is just another name for "Tangent."
The Two Must-Know Identities
1. The Quotient Identity:
\( \tan \theta \equiv \frac{\sin \theta}{\cos \theta} \)
Use this to turn a tan into sines and cosines to make equations easier to solve.
2. The Pythagorean Identity:
\( \sin^2 \theta + \cos^2 \theta \equiv 1 \)
This is just Pythagoras' Theorem hidden inside a circle! It is incredibly useful for swapping between \( \sin^2 \) and \( \cos^2 \).
Common Mistake: Be careful with notation! \( \sin^2 \theta \) means \( (\sin \theta)^2 \). It does not mean \( \sin(\theta^2) \).
Quick Takeaway: If you see a mix of \( \sin^2 \) and \( \cos \) in an equation, use \( \sin^2 \theta = 1 - \cos^2 \theta \) to make everything the same.
4. Solving Trigonometric Equations
Solving these is like solving a puzzle. The goal is to get the trig function by itself, then find the angles.
Step-by-Step Guide:
Step 1: Simplify. Use your identities to get the equation in terms of just one function (e.g., only \( \sin \)).
Step 2: Isolate. Get the function on its own (e.g., \( \cos x = \frac{1}{2} \)).
Step 3: Inverse. Use your calculator (\( \cos^{-1} \)) to find the first angle.
Step 4: Find the partners. Use the graph symmetries or a CAST diagram to find all other angles in the range.
Step 5: Check the range. Make sure all your answers fit within the limits given in the question (like \( 0 \) to \( 360 \)).
Quadratic Trig Equations
Sometimes you’ll see something like \( 2\sin^2 x - \sin x - 1 = 0 \).
Don't panic! This is just a quadratic equation in disguise. If you replace \( \sin x \) with \( y \), it becomes \( 2y^2 - y - 1 = 0 \). Solve for \( y \), then set your answers back to \( \sin x \) and solve for the angles.
Quick Review Box:
- Is the angle in degrees or radians? (Check your calculator mode!)
- Did I find all the solutions in the range?
- Did I use \( \sin^2 \theta + \cos^2 \theta = 1 \) if there are squares involved?
Key Takeaway: Treat the trig function like a single variable (like \( x \)) until the very last step when you need to find the angle.
Summary of Trigonometry
Trigonometry at AS Level is all about moving from "triangles" to "functions." Remember that the Sine and Cosine rules work for any triangle, graphs repeat every \( 360^\circ \) (or \( 180^\circ \) for tan), and identities are your best friends for simplifying difficult equations. Keep practicing those graph sketches—they are the best way to visualize your answers!