Welcome to the World of Trigonometry!
Welcome to one of the most useful and fascinating chapters in Pure Maths! You might remember Trigonometry from earlier years as "just stuff to do with right-angled triangles," but at the International AS Level, we take it much further. We will explore how triangles work (even when they don't have a 90° angle), learn about a new way to measure angles called radians, and look at the beautiful, repeating patterns of trigonometric graphs.
Don't worry if this seems a bit abstract at first. Think of Trigonometry as the "language of cycles"—it helps us describe everything from the way sound travels to how the tides rise and fall. Let’s dive in!
1. Measuring Angles: Degrees vs. Radians
Most of us are used to measuring angles in degrees (where 360° is a full circle). However, in advanced maths, we often use radians.
What is a Radian?
Imagine taking the radius of a circle and "wrapping" it around the edge (the circumference). The angle created when the arc length equals the radius is exactly 1 radian.
The Golden Rule of Conversion:
\(180^{\circ} = \pi \text{ radians}\)
To convert, use these simple steps:
1. Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
2. Radians to Degrees: Multiply by \(\frac{180}{\pi}\)
Memory Aid: Think of radians as the "natural" measurement for circles. If you see a \(\pi\) in an angle, it’s almost certainly in radians!
Quick Review: Common Angles
- \(90^{\circ} = \frac{\pi}{2}\) rad
- \(60^{\circ} = \frac{\pi}{3}\) rad
- \(45^{\circ} = \frac{\pi}{4}\) rad
- \(30^{\circ} = \frac{\pi}{6}\) rad
Key Takeaway: Always check your calculator mode! If the question uses \(\pi\) or "rad," make sure your calculator is in RAD mode. If it uses °, stay in DEG mode.
2. Arcs and Sectors
When we work in radians, the formulas for the "crust" of a pizza (the arc) and the "slice" of the pizza (the sector) become much simpler.
Arc Length (\(l\))
The distance along the curved edge of a sector is:
\(l = r\theta\)
Area of a Sector (\(A\))
The area of the "slice" is:
\(A = \frac{1}{2}r^{2}\theta\)
Note: In both these formulas, \(\theta\) must be in radians.
Common Mistake to Avoid: Don't use these formulas with degrees! If a question gives you 60°, convert it to \(\frac{\pi}{3}\) before plugging it into \(r\theta\).
3. The Sine and Cosine Rules
These rules allow us to solve any triangle, not just right-angled ones. For a triangle with sides \(a, b, c\) and opposite angles \(A, B, C\):
The Sine Rule
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
When to use it: Use this when you have "matching pairs" (an angle and its opposite side).
The Cosine Rule
\(a^{2} = b^{2} + c^{2} - 2bc \cos A\)
When to use it: Use this when you have two sides and the angle between them (SAS), or when you have all three sides (SSS).
Area of any Triangle
\(\text{Area} = \frac{1}{2}ab \sin C\)
Analogy: Think of this as the "Side-Angle-Side" area formula. You just need two sides and the "filling" (the angle between them).
Key Takeaway: If you are stuck on a triangle problem, check if you have a "matching pair." If yes, use the Sine Rule. If no, use the Cosine Rule!
4. Trig Functions and their Graphs
Trigonometric functions repeat themselves over and over. This is called periodicity.
- \(y = \sin \theta\): Starts at (0,0), goes up to 1, down to -1. Repeats every 360° (\(2\pi\)).
- \(y = \cos \theta\): Starts at (0,1), goes down to -1, back up to 1. Repeats every 360° (\(2\pi\)).
- \(y = \tan \theta\): Has vertical "walls" (called asymptotes) at 90°, 270°, etc., because you can't divide by zero! Repeats every 180° (\(\pi\)).
Did you know? The sine wave is the shape of a pure musical note. If you look at a sound wave on an oscilloscope, you're looking at trigonometry in action!
5. Trigonometric Identities
Identities are mathematical facts that are always true. They are like tools in a toolbox that help you simplify complicated equations.
Identity 1:
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
Identity 2:
\(\sin^{2} \theta + \cos^{2} \theta = 1\)
Encouraging Note: This second one is just Pythagoras' Theorem in disguise! On a circle with radius 1, the height is \(\sin \theta\) and the base is \(\cos \theta\), so \(a^2 + b^2 = c^2\) becomes \(\sin^2 + \cos^2 = 1^2\).
Key Takeaway: If you see a \(\tan \theta\) in an equation, try replacing it with \(\frac{\sin \theta}{\cos \theta}\). If you see \(\sin^2 \theta\), try replacing it with \(1 - \cos^2 \theta\).
6. Solving Trigonometric Equations
Solving an equation like \(\sin \theta = 0.5\) is a bit like a treasure hunt. There is usually more than one answer!
Step-by-Step Guide:
- Find the Principal Value: Use your calculator (e.g., \(\sin^{-1}(0.5)\)). This gives you your first answer (\(30^{\circ}\)).
- Find the Second Value: Use the symmetry of the graphs or a CAST diagram.
- For Sine: Second value is \(180^{\circ} - \text{PV}\)
- For Cosine: Second value is \(360^{\circ} - \text{PV}\)
- For Tangent: Second value is \(180^{\circ} + \text{PV}\)
- Check the Range: Add or subtract the period (360° for sin/cos, 180° for tan) to find other values within the range requested in the question (e.g., \(0 \le \theta \le 360\)).
Common Mistake: Students often stop after finding one answer on their calculator. Always look for the second (or third!) value in the range.
Summary of Key Points:
- Radians: The standard for calculus and circles (\(\pi = 180^{\circ}\)).
- Sectors: \(l = r\theta\) and \(A = \frac{1}{2}r^2\theta\).
- Rules: Sine rule for pairs; Cosine rule for SAS or SSS.
- Identities: Use \(\sin^2 + \cos^2 = 1\) to switch between sine and cosine.
- Equations: Always look for multiple solutions using graph symmetry.
Don't worry if this seems tricky at first—Trigonometry is all about practice. Once you see the patterns in the graphs, everything else starts to click!