Right-angled trigonometry, sine rule and cosine rule

Introduction to Trigonometry

Welcome to one of the most practical chapters in IB Mathematics! Trigonometry is the study of the relationships between the sides and angles of triangles. Whether you are aiming to be an architect, a pilot, or a video game designer, these tools are essential. We will start with the basics of right-angled triangles and then move on to the Sine Rule and Cosine Rule, which allow us to solve for any triangle, even if it doesn't have a right angle.

Quick Note: Before you start, always check your calculator mode! For this chapter, you will usually need to be in Degree mode unless the question specifically mentions radians (which are covered in the "Radian measure" chapter).

1. Right-Angled Trigonometry

If a triangle has a \(90^\circ\) angle, we can use Pythagoras’ Theorem and the primary trigonometric ratios.

Pythagoras’ Theorem

Used to find a missing side when you already know two other sides:
\(a^2 + b^2 = c^2\)
Where \(c\) is always the hypotenuse (the longest side, opposite the right angle).

SOH CAH TOA

This famous mnemonic helps you remember the ratios for an angle \(\theta\):

SOH: \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
CAH: \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
TOA: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Example: If you know the angle and the hypotenuse, and you want to find the side opposite the angle, use SOH: \( \text{Opposite} = \text{Hypotenuse} \times \sin \theta \).

Angles of Elevation and Depression

In word problems, you will often see these terms:

1. Angle of Elevation: The angle looking up from the horizontal line.
2. Angle of Depression: The angle looking down from the horizontal line.

Don't worry if this seems tricky: A common mistake is to put the angle of depression inside the triangle near the vertical line. Always draw a horizontal line first; the angle is measured from that line!

Key Takeaway: Right-angled trig only works for triangles with a \(90^\circ\) angle. For everything else, we need the rules below.

2. The Sine Rule

The Sine Rule works for any triangle. To use it, we label the angles with capital letters \(A, B, C\) and the sides opposite them with lowercase letters \(a, b, c\).

The Formula

\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

You can also flip the whole thing if you are looking for an angle:
\(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)

When to use it?

Use the Sine Rule when you have "opposite pairs" (an angle and its opposite side). Specifically:
- Two angles and one side (AAS or ASA).
- Two sides and a non-included angle (SSA).

The Ambiguous Case (The "Double Trouble")

This is a specific IB favorite! If you are given two sides and a non-included acute angle, there might be two possible triangles. This happens because \(\sin \theta = \sin(180^\circ - \theta)\).
For example, if your calculator says \(\theta = 40^\circ\), there is a chance the angle could actually be \(140^\circ\). Check if the second angle still leaves room for the third angle in the triangle!

Key Takeaway: If you have a matching pair (side and opposite angle), reach for the Sine Rule!

3. The Cosine Rule

The Cosine Rule is like Pythagoras’ Theorem but "upgraded" for triangles that don't have a right angle.

The Formula

To find a side:
\(a^2 = b^2 + c^2 - 2bc \cos A\)

To find an angle (rearranged):
\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

When to use it?

Use the Cosine Rule when you don't have enough "opposite pairs" for the Sine Rule:
- SAS: Two sides and the included angle (the angle between the two sides).
- SSS: All three sides and you need to find an angle.

Did you know? If angle \(A\) is \(90^\circ\), then \(\cos 90^\circ = 0\). The formula becomes \(a^2 = b^2 + c^2\), which is just Pythagoras!

Key Takeaway: The Cosine Rule is your "Plan B" when the Sine Rule doesn't work because you lack a side-angle pair.

4. Area of a Triangle

Forget \( \frac{1}{2} \times \text{base} \times \text{height} \) for a moment. In trigonometry, we can find the area using two sides and the angle between them.

The Formula

\(\text{Area} = \frac{1}{2}ab \sin C\)

Important Tip: The angle \(C\) must be the "included angle"—the one tucked between sides \(a\) and \(b\). If you have the wrong angle, use the Sine or Cosine rule first to find the one you need!

5. Problem-Solving Strategy

When faced with a complex geometry problem, follow these steps:

1. Draw a diagram: If one isn't provided, sketch it! Label all given sides and angles.
2. Identify the triangle: Is it right-angled? If yes, use SOH CAH TOA. If no, move to step 3.
3. Choose your rule:
- Have a side-angle pair? Use Sine Rule.
- Have two sides and the included angle? Use Cosine Rule.
- Have three sides? Use Cosine Rule.
4. Check for "The Ambiguous Case": Only if you used the Sine Rule to find an angle.
5. Rounding: In the IB, unless stated otherwise, give your final answer to 3 significant figures, but use exact values during your intermediate steps to avoid rounding errors!

Common Mistake to Avoid: Ensure your calculator isn't in Radian (Rad) mode when you are working with degrees. This is the most common reason for getting "weird" negative numbers for side lengths!

Summary of Key Formulae

Pythagoras: \(a^2 + b^2 = c^2\) (Right-angled only)
Trig Ratios: \(\sin \theta = \frac{O}{H}\), \(\cos \theta = \frac{A}{H}\), \(\tan \theta = \frac{O}{A}\) (Right-angled only)
Sine Rule: \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
Cosine Rule: \(a^2 = b^2 + c^2 - 2bc \cos A\)
Area: \(\text{Area} = \frac{1}{2}ab \sin C\)