Mathematics (Specification B) | Trigonometry

Welcome to the World of Trigonometry!

Hello future mathematician! This chapter, Trigonometry (often called "Trig"), is one of the most practical and exciting parts of geometry. Don't worry if the long names sound intimidating; we're essentially learning how to find missing sides and angles in triangles using simple ratios.

Trigonometry is used by surveyors, architects, navigators, and even video game designers! Ready to jump in and learn how to measure the height of a skyscraper without climbing it? Let's go!

1. Right-Angled Trigonometry: SOH CAH TOA

Trigonometry starts with the right-angled triangle (a triangle containing a 90° angle). Before we can use any formula, we need to correctly label the sides relative to the angle we are interested in (\(\theta\)).

Labeling the Triangle

Every side has a specific name:

• Hypotenuse (H): Always the longest side, and always opposite the right angle (90°).
• Opposite (O): The side directly across from the angle \(\theta\) that you are using.
• Adjacent (A): The side next to (adjacent to) the angle \(\theta\). (It helps form the angle, but isn't the hypotenuse).

Tip: If you move \(\theta\), the Opposite and Adjacent sides swap positions! The Hypotenuse stays the same.

The Three Key Ratios (SOH CAH TOA)

These three ratios link the angles of a right-angled triangle to the lengths of its sides.

The essential mnemonic to remember is: SOH CAH TOA

SOH	Sine \(\theta\)	=	Opposite / Hypotenuse	\(\sin \theta = \frac{O}{H}\)
CAH	Cosine \(\theta\)	=	Adjacent / Hypotenuse	\(\cos \theta = \frac{A}{H}\)
TOA	Tangent \(\theta\)	=	Opposite / Adjacent	\(\tan \theta = \frac{O}{A}\)

Step-by-Step: Finding a Missing Side

Follow these three steps every time:

1. Label: Label the sides O, A, and H based on the given angle.
2. Select: Choose the ratio (SOH, CAH, or TOA) that involves the side you know and the side you want to find.
3. Calculate: Rearrange the equation and solve for the unknown side.

Example: If you know the Opposite side (O) and want to find the Hypotenuse (H), you must use SOH (Sine).

Step-by-Step: Finding a Missing Angle

If you know two sides and want to find a missing angle (\(\theta\)):

1. Label: Label the sides O, A, and H relative to the missing angle \(\theta\).
2. Select: Choose the ratio (SOH, CAH, or TOA) that involves the two sides you know.
3. Use the Inverse Function: To isolate \(\theta\), you must use the inverse trig functions on your calculator: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).

Common Mistake Alert: Students often forget the inverse button! If you are finding an ANGLE, you MUST use the Shift/Second function button on your calculator (e.g., \(\cos^{-1}\)).

2. Angles of Elevation and Depression

These are essential terms for practical trigonometry problems (like finding the height of a flagpole). Both angles are measured from the horizontal line of sight.

Angle of Elevation

Imagine you are looking straight ahead (the horizontal). The Angle of Elevation is the angle measured upwards from the horizontal to look at an object (like a bird or the top of a building).

Angle of Depression

The Angle of Depression is the angle measured downwards from the horizontal to look at an object (like a boat from a cliff).

Crucial Connection: If you are standing on a cliff (A) looking at a boat (B), the angle of depression from A to B is always equal to the angle of elevation from B to A. Why? Because the horizontal line at the top and the ground are parallel, forming a Z-shape! (Alternate interior angles are equal).

3. The Sine Rule

What if the triangle doesn't have a right angle? Don't panic! We use two special rules: The Sine Rule and the Cosine Rule.

When Do I Use the Sine Rule?

The Sine Rule is used when you have a "matched pair" of information. This means you know an angle AND its directly opposite side.
You typically need to know:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA)

The Sine Rule Formula

To make this simple, we label the vertices (angles) with capital letters (A, B, C) and the sides opposite them with corresponding lowercase letters (a, b, c).

Formula to find a missing side: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Formula to find a missing angle: \[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\] Memory Aid: If you are finding a side, put the sides (a, b, c) on top. If you are finding an angle, put the sines of the angles (\(\sin A\), \(\sin B\), \(\sin C\)) on top!

Step-by-Step: Using the Sine Rule

1. Identify the Known Pair: Find the side and angle that are opposite each other and both known.
2. Set up the Equation: Write the known pair equal to the required unknown pair. (e.g., \(\frac{a}{\sin A} = \frac{b}{\sin B}\))
3. Solve: Rearrange the equation to find the unknown side or angle. If finding an angle, remember to use the inverse function (\(\sin^{-1}\)) in the final step!

4. The Cosine Rule

The Cosine Rule is your go-to when the Sine Rule fails. It is a bit longer, but very reliable!

When Do I Use the Cosine Rule?

You use the Cosine Rule in two specific scenarios (where you don't have a full angle-side pair):

• Scenario 1 (Finding a Side): You know two sides and the Included Angle (SAS). The included angle is the angle between the two known sides.
• Scenario 2 (Finding an Angle): You know all three sides (SSS).

The Cosine Rule Formulas

A. Finding a Missing Side (SAS Known)

To find side a, given sides b and c and the angle A between them: \[a^2 = b^2 + c^2 - 2bc \cos A\] Analogy: This looks similar to Pythagoras' theorem (\(a^2 = b^2 + c^2\)), but with an extra correction factor (\(- 2bc \cos A\)) to adjust for the triangle not being right-angled.

B. Finding a Missing Angle (SSS Known)

This formula is simply the first one rearranged. It is used when you know all three sides and need to find one of the angles (say, angle A). \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\] Tip for struggling students: Remember that the side you are trying to find the angle for (side \(a\)) is the one that is subtracted at the end (\(- a^2\)).

Don't forget: After calculating the value of \(\cos A\), you must use the inverse function \(\cos^{-1}\) to find the actual angle A!

5. Calculating the Area of a Triangle

You might remember the simple area formula: Area = \(\frac{1}{2} \times \text{base} \times \text{height}\). But often, in trigonometry problems, the perpendicular height is unknown.

Fortunately, there is a simple formula using Sine that works for ANY triangle, provided you know two sides and the angle included between them (SAS).

The Area Formula (Using Sine)

If you have sides a and b, and the included angle C: \[\text{Area} = \frac{1}{2} ab \sin C\]

The Golden Rule: The angle you use (C) MUST be the angle included between the two sides (a and b) you are using.

Example: If you know sides 4 cm and 5 cm, and the angle *between* them is 30°, the Area is \(\frac{1}{2} \times 4 \times 5 \times \sin 30\).