Basic trigonometry - Mathematics B (MEI) - H630 - Cambridge OCR AS Level

Welcome to Basic Trigonometry!

Trigonometry might sound like a mouthful, but it’s essentially the study of how the sides and angles of triangles relate to one another. Whether you are aiming to be an architect, a game developer, or a pilot, trigonometry is the secret tool used to navigate and build our world. In this chapter, we will refresh the basics and then stretch those ideas to cover any angle imaginable!

1. Back to Basics: Right-Angled Triangles

Before we dive into the deep end, let's make sure our foundation is solid. In a right-angled triangle, the sides are named based on their position relative to a specific angle, usually called theta (\(\theta\)).

The Three Sides:
1. Hypotenuse: The longest side, always opposite the right angle.
2. Opposite: The side across from angle \(\theta\).
3. Adjacent: The side next to angle \(\theta\) (that isn't the hypotenuse).

SOH CAH TOA: Your Best Friend

This classic mnemonic helps you remember the three primary ratios:
- SOH: \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- CAH: \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
- TOA: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Don't worry if this seems tricky at first! Just remember that these are just "recipes" for finding missing lengths or angles.

Quick Review:
- If you have an angle and one side, you can find the other sides.
- If you have two sides, you can find the angle using the "inverse" functions on your calculator (\(\sin^{-1}, \cos^{-1}, \tan^{-1}\)).

Common Mistake: Always check your calculator mode! For this chapter, make sure it is set to Degrees (D), not Radians (R).

Key Takeaway: SOH CAH TOA only works for right-angled triangles. For other triangles, we need the Sine and Cosine rules (coming up later!).

2. Trigonometry for Any Angle: The Unit Circle

What happens when an angle is bigger than \(90^\circ\)? To understand this, we use the Unit Circle—a circle with a radius of 1 centered at the origin \((0,0)\).

Imagine a point moving around this circle. The angle \(\theta\) starts from the positive x-axis and turns counter-clockwise.
- The x-coordinate of the point is \(\cos \theta\).
- The y-coordinate of the point is \(\sin \theta\).
- The gradient (slope) of the line from the center to the point is \(\tan \theta\), which is \(\frac{y}{x}\).

The "CAST" Diagram (Memory Aid)

The circle is split into four quadrants. This helps us know if a value is positive or negative:
- Quadrant 1 (\(0^\circ\) to \(90^\circ\)): All are positive.
- Quadrant 2 (\(90^\circ\) to \(180^\circ\)): Only Sin is positive.
- Quadrant 3 (\(180^\circ\) to \(270^\circ\)): Only Tan is positive.
- Quadrant 4 (\(270^\circ\) to \(360^\circ\)): Only Cos is positive.

Analogy: Think of it like a compass. Depending on which "neighborhood" you are in, only certain trig functions are "welcome" (positive).

Did you know? This is why \(\sin(150^\circ)\) is the same as \(\sin(30^\circ)\). They both have the same height (y-value) on the unit circle!

Key Takeaway: \(\sin \theta = y\), \(\cos \theta = x\), and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

3. Exact Values You Must Know

In many exam questions, you'll be asked for "exact values." This means no decimals! You should memorize these values for \(30^\circ, 45^\circ\), and \(60^\circ\).

Handy Reference:
- \(\sin(30^\circ) = \frac{1}{2}\) and \(\cos(60^\circ) = \frac{1}{2}\)
- \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\) and \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)
- \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) and \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
- \(\tan(45^\circ) = 1\)

Key Takeaway: If you see \(\sqrt{2}\) or \(\sqrt{3}\) in a question, it’s a huge hint that you should be using these exact values!

4. Graphs of Trig Functions

If you plot the values of \(\sin, \cos\), and \(\tan\) as a graph, you see beautiful repeating patterns called periodic waves.

The Sine Wave (\(y = \sin \theta\)): Starts at \((0,0)\), goes up to 1 at \(90^\circ\), and repeats every \(360^\circ\).
The Cosine Wave (\(y = \cos \theta\)): Starts at \((0,1)\), goes down to 0 at \(90^\circ\), and repeats every \(360^\circ\).
The Tangent Graph (\(y = \tan \theta\)): Looks like several separate "flicks." It has asymptotes (lines it never touches) at \(90^\circ, 270^\circ\), etc., and repeats every \(180^\circ\).

Quick Review of Transformations:
- \(y = a\sin \theta\): This is a stretch vertically (makes the wave taller).
- \(y = \sin(b\theta)\): This is a stretch horizontally (squashes or stretches the wave along the x-axis).
- \(y = \sin \theta + c\): This is a translation up or down.

Key Takeaway: Sine and Cosine repeat every \(360^\circ\). Tangent repeats every \(180^\circ\).

5. The Sine and Cosine Rules

When a triangle doesn't have a right angle, we use these two powerful rules. Label your triangle with capital letters \(A, B, C\) for angles and lowercase \(a, b, c\) for the sides opposite them.

The Sine Rule

\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Use this when you have a "matching pair" (a side and its opposite angle) plus one other piece of information.

The Cosine Rule

\(a^2 = b^2 + c^2 - 2bc \cos A\)
Use this when:
1. You have two sides and the angle between them (SAS).
2. You have all three sides and want to find an angle (SSS).

Area of Any Triangle

\(\text{Area} = \frac{1}{2}ab \sin C\)
You don't need the vertical height! Just two sides and the angle between them.

Key Takeaway: Think of the Cosine Rule as Pythagoras' Theorem with a "correction factor" (\(- 2bc \cos A\)) for non-right triangles.

6. Trigonometric Identities

Identities are equations that are always true. They are incredibly useful for simplifying complicated expressions.

1. The Tangent Identity:
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

2. The Pythagorean Identity:
\(\sin^2 \theta + \cos^2 \theta = 1\)

Note: \(\sin^2 \theta\) is just shorthand for \((\sin \theta)^2\).

How to use them: If you have an equation with both \(\sin^2 \theta\) and \(\cos \theta\), you can replace \(\sin^2 \theta\) with \((1 - \cos^2 \theta)\) so that everything is in terms of cosine. This often turns the problem into a quadratic equation!

Key Takeaway: These identities allow you to "swap" trig functions to make equations easier to solve.

7. Solving Trigonometric Equations

Solving an equation like \(\sin \theta = 0.5\) is like finding all the times a wave hits a certain height within a specific interval (like \(0^\circ\) to \(360^\circ\)).

Step-by-Step Process:

1. Isolate the trig function: Get it in the form \(\sin \theta = \dots\)
2. Find the "Principal Value": Use your calculator (\(\theta = \sin^{-1}(0.5) = 30^\circ\)).
3. Find other solutions: Use the symmetry of the graphs or the CAST diagram.
- For \(\sin \theta\): Second solution is \(180^\circ - \text{PV}\).
- For \(\cos \theta\): Second solution is \(360^\circ - \text{PV}\) (or \(-\text{PV}\)).
- For \(\tan \theta\): Other solutions are \(\text{PV} + 180^\circ\), \(\text{PV} - 180^\circ\), etc.
4. Check the range: Make sure all your answers are within the limits set in the question (e.g., \(0 \le \theta \le 360\)).

Common Mistake: Forgetting the second (or third) solution! Trig equations usually have more than one answer in a full circle.

Key Takeaway: Your calculator only gives you one answer (the Principal Value). You must use your knowledge of graphs or CAST to find the others!

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