Introduction to Trigonometry

Welcome to the study of triangles! Trigonometry might sound like a complex word, but it simply means "triangle measurement." In this chapter, we will explore how to find missing sides and angles in any triangle—whether it has a perfect \(90^{\circ}\) corner or not. These skills are the "bread and butter" for architects, surveyors, and even game developers who need to calculate distances and paths in 3D space.

Don't worry if this seems tricky at first! We will break it down into two main parts: triangles with right angles and triangles without them. Once you master a few key formulas, you'll be able to solve almost any triangle problem the IB throws at you.

1. The Foundation: Pythagoras’ Theorem

Before we dive into angles, we must remember the most famous rule for right-angled triangles. Pythagoras’ Theorem allows us to find a missing side if we already know the other two sides.

For a right-angled triangle with legs \(a\) and \(b\), and the longest side (the hypotenuse) \(c\):

\(a^2 + b^2 = c^2\)

Quick Tip: The hypotenuse is always the side directly across from the \(90^{\circ}\) angle. It is always the longest side!

Common Mistake: Make sure you are only using this formula for triangles that have a \(90^{\circ}\) angle. It won't work for "wonky" triangles!

2. Right-Angled Trigonometry (SOH CAH TOA)

When we know an angle and a side (or want to find an angle), we use the trigonometric ratios: Sine, Cosine, and Tangent. To remember which is which, we use the famous mnemonic SOH CAH TOA.

First, label your sides based on the angle (\(\theta\)) you are looking at:

  • Opposite (O): The side across from the angle.
  • Adjacent (A): The side next to the angle (that isn't the hypotenuse).
  • Hypotenuse (H): The longest side across from the right angle.
The Ratios:

\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\) (SOH)

\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) (CAH)

\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\) (TOA)

Finding an Angle:

If you know the sides but need the angle, use the "inverse" functions on your calculator (usually shift/2nd + sin/cos/tan):

\(\theta = \sin^{-1}\left(\frac{O}{H}\right)\)

Key Takeaway: Always check your calculator is in Degree mode (DEG) unless the question specifically asks for radians!

3. Angles of Elevation and Depression

In real-world IB problems, you often see these terms. They always describe the angle measured from a horizontal line.

  • Angle of Elevation: The angle looking up from the horizontal.
  • Angle of Depression: The angle looking down from the horizontal.

Did you know? The angle of elevation from point A to point B is exactly the same as the angle of depression from point B back to point A. This is because they are "alternate interior angles" between parallel horizontal lines!

4. Non-Right-Angled Trigonometry

What if there is no \(90^{\circ}\) angle? SOH CAH TOA won't work here. Instead, we use two powerful rules: the Sine Rule and the Cosine Rule.

The Sine Rule

Use this when you have "matching pairs" of an angle and its opposite side.

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

(Use this version to find a side)

OR

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

(Use this version to find an angle)

When to use: When you know two angles and one side, or two sides and an angle opposite one of them.

The Cosine Rule

Use this when you don't have enough information for the Sine Rule (no "matching pairs").

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

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

When to use:
1. SAS: You know two Sides and the Angle between them.
2. SSS: You know all three Sides.

Analogy: Think of the Cosine Rule as Pythagoras' Theorem's "older sibling." If the angle \(C\) is \(90^{\circ}\), \(\cos 90^{\circ} = 0\), and the formula turns back into \(c^2 = a^2 + b^2\)!

5. Area of a Triangle

Forget \(Base \times Height \div 2\) for a moment. In trigonometry, we can find the area of any triangle if we know two sides and the included angle (SAS).

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

Make sure angle \(C\) is the one tucked between sides \(a\) and \(b\).

6. The Ambiguous Case of the Sine Rule (HL Only)

For Higher Level (HL) students, there is a special situation called the Ambiguous Case (SSA). If you are given two sides and an acute angle not between them, it is possible that two different triangles could be formed.

When you use \(\sin^{-1}\) to find an angle, your calculator gives you an acute angle (\(\theta_1\)). However, there might be an obtuse version (\(\theta_2\)) that also works!

\(\theta_2 = 180^{\circ} - \theta_1\)

Quick Check: If \((180^{\circ} - \theta_1) + (\text{given angle}) < 180^{\circ}\), then a second triangle exists!

Step-by-Step: Which Rule to Use?

  1. Is there a right angle?
    • Yes \(\implies\) Use Pythagoras or SOH CAH TOA.
    • No \(\implies\) Go to step 2.
  2. Do I have a "matching pair" (side and opposite angle)?
    • Yes \(\implies\) Use the Sine Rule.
    • No \(\implies\) Use the Cosine Rule.
  3. Am I looking for Area?
    • Use \(Area = \frac{1}{2} ab \sin C\).

Key Summary Table

Topic: Right-Angled Trig
Tools: \(a^2 + b^2 = c^2\), \(\sin = \frac{O}{H}\), \(\cos = \frac{A}{H}\), \(\tan = \frac{O}{A}\)

Topic: Non-Right-Angled Trig
Tools: Sine Rule (\(2\) sides, \(2\) angles), Cosine Rule (SAS or SSS)

Topic: Triangle Area
Tools: \(\frac{1}{2} ab \sin C\)

Final Encouragement: Trigonometry is all about identifying the "pattern" of the information you are given. Once you label your triangle sides \(a, b, c\) and angles \(A, B, C\), the formulas do the heavy lifting for you!