Welcome to the World of Trigonometry!
Trigonometry might sound like a mouthful, but it is essentially the study of how the sides and angles of triangles relate to each other. Whether you are aiming to be an architect, a game developer, or an engineer, trigonometry is the "secret sauce" used to calculate distances and model waves. In this PP1: Pure Maths section, we will move beyond basic right-angled triangles and explore how trig works for any triangle and even on a graph!
1. Measuring Angles: Degrees vs. Radians
Since primary school, you have used degrees (where \( 360^\circ \) is a full circle). However, in A-Level Maths, we use a more "natural" unit called the radian.
What is a Radian?
Imagine taking the radius of a circle and bending it along the edge (the arc). The angle created at the center is exactly 1 radian. Because the circumference of a circle is \( 2\pi r \), there are exactly \( 2\pi \) radians in a full circle.
The Golden Conversion:
\( 180^\circ = \pi \text{ radians} \)
How to convert:
1. Degrees to Radians: Multiply by \( \frac{\pi}{180} \)
2. Radians to Degrees: Multiply by \( \frac{180}{\pi} \)
Memory Trick: To get radians, you usually want a \(\pi\) in your answer, so put \(\pi\) on top of the fraction!
Key Takeaway:
• Always check if your calculator is in DEG or RAD mode before starting a calculation! This is the most common mistake students make.
2. Arcs and Sectors
Using radians makes calculating the length of a "crust" of a pizza (arc length) or the area of a "slice" (sector area) much simpler.
Arc Length (\(l\)):
\( l = r\theta \)
(Where \(r\) is the radius and \(\theta\) is the angle in radians).
Area of a Sector (\(A\)):
\( A = \frac{1}{2}r^2\theta \)
Analogy: Think of \(\theta\) as the percentage of the circle you are using. In degrees, the formulas are messy. In radians, they are "clean" because radians are built directly from the radius.
3. The Sine and Cosine Rules
We use these rules when a triangle does not have a right angle. In these formulas, capital letters (\(A, B, C\)) are angles, and lowercase letters (\(a, b, c\)) are the sides opposite those angles.
The Sine Rule
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
When to use it: When you have a "matching pair" (a side and its opposite angle) plus one other piece of information.
The Cosine Rule
\( a^2 = b^2 + c^2 - 2bc \cos A \)
or rearranged for the angle:
\( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
When to use it: When you have SSS (Side-Side-Side) or SAS (Side-Angle-Side).
Area of Any Triangle
\( \text{Area} = \frac{1}{2}ab \sin C \)
Don't worry if this seems tricky at first! Just remember: to use this area formula, you need two sides and the "included" angle (the angle sandwiched between them).
4. Trigonometric Graphs and Properties
Trig functions repeat themselves—this is called periodicity. Understanding the shapes of these graphs helps you find multiple solutions to equations.
1. Sine Graph \(y = \sin \theta\): Starts at (0,0), goes up to 1, down to -1. Repeats every \( 360^\circ \) (or \( 2\pi \) radians).
2. Cosine Graph \(y = \cos \theta\): Starts at (0,1). It's just the sine graph shifted left by \( 90^\circ \). Repeats every \( 360^\circ \) (or \( 2\pi \) radians).
3. Tangent Graph \(y = \tan \theta\): Looks like several "snakes" climbing up. It has asymptotes (lines it never touches) at \( 90^\circ, 270^\circ \), etc. Repeats every \( 180^\circ \) (or \( \pi \) radians).
Did you know?
Because these graphs repeat, an equation like \( \sin \theta = 0.5 \) has an infinite number of answers! We usually only look for answers within a specific range, like \( 0 \le \theta \le 360^\circ \).
5. Essential Trigonometric Identities
Identities are mathematical facts that are always true. You will use these to simplify complex expressions or solve equations.
Identity 1:
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
Identity 2 (The Pythagorean Identity):
\( \sin^2 \theta + \cos^2 \theta = 1 \)
Quick Review Box:
• If you see \( \sin^2 \theta \) and \( \cos \theta \) in the same equation, use Identity 2 to turn everything into Cosine!
• Remember: \( \sin^2 \theta \) is just shorthand for \( (\sin \theta)^2 \).
6. Solving Trigonometric Equations
This is where everything comes together. To solve an equation like \( 2\sin^2 \theta + 5\cos \theta = 4 \):
Step-by-Step Process:
1. Harmonize: Use identities to make all trig terms the same (e.g., change \( \sin^2 \theta \) to \( 1 - \cos^2 \theta \)).
2. Solve for the function: This often creates a quadratic equation (e.g., \( 2\cos^2 \theta - 5\cos \theta + 2 = 0 \)). Let \( x = \cos \theta \) to make it easier to see.
3. Find the "Principal Value": Use your calculator (\( \cos^{-1} \)) to find the first angle.
4. Find Other Values: Use the symmetry of the graph or a CAST diagram to find all other angles in the required range.
Common Mistake to Avoid:
When you solve \( \cos^2 \theta = 0.25 \), remember that \( \cos \theta \) can be +0.5 OR -0.5. Don't forget the negative square root, or you'll miss half your answers!
Key Takeaway:
• Trigonometry is all about patterns. If you know the basic graph shape and your two main identities, you can solve almost any problem in this section.