Welcome to the World of Trigonometry!
Welcome! Trigonometry might sound like a big, scary word, but it really just means "triangle measurement." In this chapter of Unit P1, we are going to explore how angles and sides relate to each other, not just in right-angled triangles, but in any triangle and even in circular shapes!
Whether you are calculating the height of a mountain or the path of a satellite, trigonometry is the tool you need. Don't worry if it seems tricky at first—we will break it down piece by piece.
1. Radian Measure: A New Way to Measure Angles
Up until now, you’ve probably measured angles in degrees (like \(90^\circ\) or \(180^\circ\)). In Advanced 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 a full circle has a circumference of \(2\pi r\), there are \(2\pi\) radians in a full circle.
The Golden Rule for Conversion
To switch between degrees and radians, just remember this simple connection:
\(180^\circ = \pi\) radians
To change Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
Example: \(60^\circ \times \frac{\pi}{180} = \frac{\pi}{3}\) radians.
To change Radians to Degrees: Multiply by \(\frac{180}{\pi}\)
Example: \(\frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ\).
Did you know? Most scientific calculators have a "DRG" button or a mode setting. Always check your calculator is in the right mode (Deg or Rad) before starting a problem!
Key Takeaway: Radians are just another language for angles. \(\pi\) is your best friend here!
2. Arc Length and Sector Area
When we use radians, the formulas for the parts of a circle become much simpler.
Arc Length
The arc length (\(s\)) is the distance along the curved edge of a "pizza slice."
Formula: \(s = r\theta\)
(Where \(r\) is the radius and \(\theta\) is the angle in radians).
Area of a Sector
The sector is the entire "slice" of the circle.
Formula: \(A = \frac{1}{2}r^2\theta\)
Common Mistake: Using degrees in these formulas! If the question gives you \(30^\circ\), you must convert it to \(\frac{\pi}{6}\) radians first, or the formula won't work.
3. The Sine and Cosine Rules
These rules allow us to find missing sides and angles in any triangle, not just right-angled ones. Label your triangle with capital letters for angles (\(A, B, C\)) and lowercase letters for the sides opposite them (\(a, b, c\)).
The Sine Rule
Use this when you have "matching pairs" (a side and its opposite angle).
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
The Cosine Rule
Use this when you have Side-Angle-Side (SAS) or Side-Side-Side (SSS).
\(a^2 = b^2 + c^2 - 2bc \cos A\)
The Area of Any Triangle
Forget "half base times height" for a moment! If you know two sides and the angle between them:
Area = \(\frac{1}{2}ab \sin C\)
The "Ambiguous Case" of the Sine Rule
Sometimes, the Sine Rule can be a bit cheeky. If you are given two sides and an acute angle that is not between them, there might be two possible triangles. This happens because \(\sin \theta\) is the same as \(\sin(180 - \theta)\).
Example: If your calculator says an angle is \(40^\circ\), the other possible answer is \(180 - 40 = 140^\circ\). Always check if the second angle fits in the triangle!
Quick Review: Pairs? Use Sine Rule. SAS or SSS? Use Cosine Rule.
4. Trigonometric Graphs
Trig functions aren't just for triangles; they are periodic, meaning they repeat forever in waves.
The Sine Wave: \(y = \sin x\)
- Starts at \((0,0)\).
- Goes up to \(1\) and down to \(-1\).
- Repeats every \(360^\circ\) (or \(2\pi\) radians).
The Cosine Wave: \(y = \cos x\)
- Starts at \((0,1)\).
- Looks exactly like the sine wave, but shifted \(90^\circ\) to the left.
- Repeats every \(360^\circ\).
The Tangent Graph: \(y = \tan x\)
- This one is the "rebel." It doesn't have a wave; it has separate branches.
- It has asymptotes (lines it never touches) at \(90^\circ, 270^\circ\), etc.
- Repeats more often: every \(180^\circ\) (or \(\pi\) radians).
Memory Aid: Think of Sine as starting at the Source \((0,0)\) and Cosine as starting at the Ceiling \((0,1)\).
5. Transforming Trig Graphs
Just like any other function, we can stretch or shift these waves.
1. \(y = a f(x)\) (Vertical Stretch): This changes the height (amplitude). \(y = 3 \sin x\) goes up to \(3\) and down to \(-3\).
2. \(y = f(ax)\) (Horizontal Stretch): This squashes or stretches the wave horizontally. \(y = \sin 2x\) makes the wave repeat twice as fast.
3. \(y = f(x) + a\) (Vertical Shift): Moves the whole graph up or down.
4. \(y = f(x + a)\) (Horizontal Shift): Moves the graph left or right. Remember: \(+ a\) moves it left, \(- a\) moves it right!
Step-by-Step for Sketching:
1. Draw the basic shape (Sine, Cos, or Tan).
2. Apply the stretch (make it taller or thinner).
3. Move the graph (up/down/left/right).
4. Mark your key points (intercepts and peaks).
Summary and Key Takeaways
- Radians: Essential for circle formulas. \(180^\circ = \pi\).
- Circle Parts: Arc is \(r\theta\), Area is \(\frac{1}{2}r^2\theta\).
- Sine/Cosine Rules: Your tools for solving any triangle. Watch out for the ambiguous case!
- Graphs: Waves that repeat. Learn their starting points and symmetries.
- Transformations: Practice moving the waves to see how the equations change the shape.
Trigonometry is all about patterns. Once you recognize the shape of the wave or the "pair" in a triangle, the rest is just using your calculator correctly. You've got this!