Welcome to the World of Trigonometry!
Welcome! Trigonometry might sound like a big, scary word, but it’s simply the study of the relationships between the sides and angles of triangles. Whether you're interested in architecture, video game design, or navigation, trigonometry is the tool that makes it all possible. In these notes, we will break down the essential rules and graphs you need for your Pearson Edexcel International AS Level (P1 and P2) exams. Let's dive in!
1. Working with Non-Right-Angled Triangles
In your earlier years, you likely used SOH CAH TOA for right-angled triangles. But what if the triangle doesn't have a 90-degree angle? That’s where the Sine Rule and Cosine Rule come to the rescue.
The Sine Rule
Use this when you have "pairs" of sides and opposite angles. The formula is:
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
Don't forget the Ambiguous Case: Sometimes, when you are given two sides and a non-included angle, there could be two possible triangles. This happens because \( \sin \theta = \sin(180^\circ - \theta) \). Always check if a second, obtuse angle could fit in your triangle!
The Cosine Rule
Use this when you have "SAS" (Side-Angle-Side) or "SSS" (Side-Side-Side). It’s like a more advanced version of Pythagoras' Theorem:
\( a^2 = b^2 + c^2 - 2bc \cos A \)
To find an angle, you can rearrange it to: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
Area of a Triangle
If you don't know the vertical height, you can find the area using two sides and the angle between them:
Area = \( \frac{1}{2} ab \sin C \)
Key Takeaway: Use the Sine Rule for pairs and the Cosine Rule when you have a "side-angle-side" sandwich or all three sides.
2. Radian Measure
In AS Level, we move beyond just degrees and start using Radians. Think of radians as a different "language" for measuring angles, just like centimeters and inches both measure length.
What is a Radian?
One radian is the angle created when the arc length of a circle is equal to its radius.
The Golden Rule: \( 180^\circ = \pi \text{ radians} \)
Arc Length and Sector Area
Using radians actually makes our formulas much simpler!
- Arc Length (s): \( s = r\theta \)
- Area of a Sector (A): \( A = \frac{1}{2} r^2 \theta \)
Note: For these formulas to work, \(\theta\) must be in radians.
Quick Tip: To convert degrees to radians, multiply by \( \frac{\pi}{180} \). To convert radians to degrees, multiply by \( \frac{180}{\pi} \).
Key Takeaway: Always check your calculator mode! If the question uses \(\pi\) or radians, make sure your calculator shows a little 'R' at the top, not a 'D'.
3. Trigonometric Graphs and Transformations
Visualizing Sine, Cosine, and Tangent as waves helps us understand how they behave over time.
The Shapes
- y = sin x: Starts at (0,0), goes up to 1, down to -1. It repeats every \( 360^\circ \) (or \( 2\pi \)).
- y = cos x: Starts at (0,1), goes down to -1 and back up. It’s just the sine wave shifted to the left!
- y = tan x: Looks like a series of "S" shapes. It has asymptotes (lines it never touches) at \( 90^\circ, 270^\circ \), etc. It repeats every \( 180^\circ \) (or \( \pi \)).
Transforming the Graphs
You need to know how changing the equation changes the graph:
- y = af(x): Vertical stretch (e.g., \( y = 3\sin x \) goes up to 3 and down to -3).
- y = f(x) + a: Vertical shift (moves the whole graph up or down).
- y = f(x + a): Horizontal shift (moves left or right). Remember: \( +a \) moves it left!
- y = f(ax): Horizontal stretch/squash (e.g., \( y = \sin 2x \) makes the wave repeat twice as often).
Key Takeaway: The period is how long the graph takes to repeat. For \(\sin x\) and \(\cos x\), it’s \( 360^\circ \). For \(\tan x\), it’s \( 180^\circ \).
4. Trigonometric Identities
Identities are mathematical facts that are true for all values. They are like "conversion tools" that let you swap one expression for another to solve tricky equations.
The Big Two
- \( \tan \theta \equiv \frac{\sin \theta}{\cos \theta} \)
- \( \sin^2 \theta + \cos^2 \theta \equiv 1 \)
Did you know? The second identity is actually just Pythagoras' Theorem (\( a^2 + b^2 = c^2 \)) hiding inside a circle with a radius of 1!
Common Mistake: Students often forget that \( \sin^2 \theta \) means \( (\sin \theta)^2 \). It does not mean \( \sin(\theta^2) \).
Key Takeaway: If an equation has both \(\sin^2 \theta\) and \(\cos \theta\), use the second identity to turn everything into \(\cos\) so you can solve it like a quadratic equation.
5. Solving Trigonometric Equations
Solving \( \sin x = 0.5 \) is different from normal algebra because there isn't just one answer—there are infinitely many! We usually solve for a specific interval (like \( 0^\circ \leq x \leq 360^\circ \)).
The Step-by-Step Process
- Simplify: Get the trig function (like \(\sin x\) or \(\cos \theta\)) by itself on one side.
- Principal Value: Use your calculator (\( \sin^{-1} \), etc.) to find the first answer.
- Find Others: Use the unit circle (CAST diagram) or the trig graphs to find the other values in the range.
- Adjust for Transformations: If the question is \( \cos(2x) = 0.5 \), solve for \( 2x \) first, find all values in a wider range, and then divide by 2 at the very end.
Example: Solve \( 6\cos^2 x + \sin x - 5 = 0 \)
Don't worry if this seems tricky! Just follow the tools:
1. Replace \( \cos^2 x \) with \( (1 - \sin^2 x) \).
2. You now have a quadratic: \( 6(1 - \sin^2 x) + \sin x - 5 = 0 \).
3. Simplify to \( 6\sin^2 x - \sin x - 1 = 0 \).
4. Factorize and solve for \( \sin x \), then find the angles \( x \).
Key Takeaway: Always check your final answers to see if they fall within the interval given in the question (e.g., \( 0 < x < 2\pi \)).
Quick Review Checklist
- Are you in Degrees or Radians mode?
- Did you check for the ambiguous case in the Sine Rule?
- When solving equations, did you find all the values in the range?
- Can you sketch the basic \( \sin, \cos, \tan \) graphs from memory?
You've got this! Trigonometry takes practice, but once you recognize the patterns, it becomes one of the most predictable and rewarding parts of Pure Mathematics.