Welcome to the World of Trigonometry!

Hey there! Get ready to explore a super useful branch of maths called Trigonometry. It sounds complicated, but it's really just the study of triangles.

Ever wondered how people measure the height of a giant tree or a tall building without climbing it? Or how video game designers create amazing 3D worlds? They use trigonometry! In this chapter, you'll learn the secrets to measuring angles and distances you can't reach, and you'll see how triangles are the hidden building blocks for so many things in the world around us.

Don't worry if it seems tricky at first. We'll break everything down into simple, easy-to-understand steps. Let's get started!


Section 1: The Basics - Meet the Right-Angled Triangle

First, a Quick Review

Trigonometry is all about a special kind of triangle: the right-angled triangle. This is any triangle that has one angle measuring exactly 90 degrees (like the corner of a square).

Naming the Sides

In trigonometry, the three sides of a right-angled triangle have special names. But here's the trick: two of the names depend on which angle you're looking from! Let's use the Greek letter θ (theta) to represent our angle.

1. The Hypotenuse
This is the easy one! The Hypotenuse is always the longest side, and it's always the side opposite the 90° right angle.

2. The Opposite Side
The Opposite side is the side directly across from the angle θ you're interested in.

3. The Adjacent Side
The Adjacent side is the side that is next to the angle θ, but isn't the hypotenuse. Think of 'adjacent' as meaning 'neighbouring'.

Example: Imagine you are standing at angle θ. The wall across the room is 'Opposite' you. The wall right next to you is 'Adjacent' to you! The slanted ceiling above is the 'Hypotenuse'.

Important: If you change your angle, the Opposite and Adjacent sides will switch places! The Hypotenuse always stays the same.

Key Takeaway

In a right-angled triangle, we must correctly label the Hypotenuse, Opposite, and Adjacent sides relative to our chosen angle (θ). This is the most important first step!


Section 2: The Three Core Ratios - SOH CAH TOA

What are Trig Ratios?

Trigonometry gives us three amazing tools that connect the angles of a right-angled triangle to the lengths of its sides. These tools are called sine, cosine, and tangent. We often shorten them to sin, cos, and tan.

The Magic Memory Aid: SOH CAH TOA

This is the most famous mnemonic in all of maths, and it's your key to success! Just say it out loud: "SOH - CAH - TOA".

Here's what it means:

SOH stands for: Sine = Opposite / Hypotenuse
$$sin(θ) = \frac{Opposite}{Hypotenuse}$$
CAH stands for: Cosine = Adjacent / Hypotenuse
$$cos(θ) = \frac{Adjacent}{Hypotenuse}$$
TOA stands for: Tangent = Opposite / Adjacent
$$tan(θ) = \frac{Opposite}{Adjacent}$$

How to Use SOH CAH TOA

When you need to find a missing side or angle, follow these simple steps:

Step 1: Pick your angle (θ). This will be either the angle you know or the angle you want to find.

Step 2: Label the three sides of the triangle: Hypotenuse (H), Opposite (O), and Adjacent (A) from the perspective of your angle θ.

Step 3: Look at what you have and what you need. Do you have O and H? Use SOH! Do you have A and H? Use CAH! Do you have O and A? Use TOA!

Step 4: Write down the equation and solve for the unknown value. You'll need a calculator for this part! Make sure it's in "Degrees" mode (look for DEG on the screen).

Common Mistakes to Avoid

- Forgetting to label the sides first. This is the #1 cause of errors!
- Mixing up which ratio to use. Always write down SOH CAH TOA to help you choose.
- When finding an angle, remember to use the inverse buttons on your calculator, which look like $$sin^{-1}, cos^{-1},$$ or $$tan^{-1}$$.

Key Takeaway

SOH CAH TOA is the key to trigonometry. It tells you which ratio (sin, cos, or tan) to use based on the sides you know and the sides you want to find.


Section 3: The Special Angles (30°, 45°, 60°)

No Calculator Needed!

Some angles are so common in maths and design that it's useful to know their exact trig values without a calculator. These special angles are 30°, 45°, and 60°. The values are simple fractions that often involve square roots.

The Special Values Table

Here are the values you should get familiar with. You can derive them from special triangles, but for now, let's focus on knowing them.

Angle (θ) | sin(θ) | cos(θ) | tan(θ)
----------------------------------------------------------------------
30° | $$ \frac{1}{2} $$ | $$ \frac{\sqrt{3}}{2} $$ | $$ \frac{1}{\sqrt{3}} $$
----------------------------------------------------------------------
45° | $$ \frac{1}{\sqrt{2}} $$ | $$ \frac{1}{\sqrt{2}} $$ | $$ 1 $$
----------------------------------------------------------------------
60° | $$ \frac{\sqrt{3}}{2} $$ | $$ \frac{1}{2} $$ | $$ \sqrt{3} $$

Did you know?

Notice a pattern? The sine of 30° is the same as the cosine of 60°. And the sine of 60° is the same as the cosine of 30°. This isn't a coincidence, and we'll see why in the next section!

Key Takeaway

Memorising or being able to quickly find the exact trig values for 30°, 45°, and 60° is a great skill that will help you solve problems more quickly and accurately.


Section 4: Important Properties and Relationships

The three trig ratios are all related to each other in cool ways. Understanding these properties (also called identities) will give you trig superpowers! These are true for any angle θ between 0° and 90°.

Property 1: The Quotient Identity

This property connects tan, sin, and cos together.
$$tan(θ) = \frac{sin(θ)}{cos(θ)}$$ This makes sense if you think about it: $$(\frac{O}{H}) / (\frac{A}{H}) = \frac{O}{H} \times \frac{H}{A} = \frac{O}{A}$$, which is the formula for tan(θ)!

Property 2: The Pythagorean Identity

This is the most famous identity, and it's named after Pythagoras' theorem.
$$sin^2(θ) + cos^2(θ) = 1$$ Note: $$sin^2(θ)$$ is just a fancy way of writing $$(sin(θ))^2$$. This rule is incredibly useful for finding sin(θ) if you know cos(θ), or vice-versa.

Property 3: Complementary Angles

Complementary angles are two angles that add up to 90°. In any right-angled triangle, the two non-right angles are always complementary. This leads to a neat trick:
$$sin(90° - θ) = cos(θ)$$ $$cos(90° - θ) = sin(θ)$$ This is why sin(30°) = cos(60°), because 30° and 60° add up to 90°!

There is also a relationship for tangent:
$$tan(90° - θ) = \frac{1}{tan(θ)}$$

Property 4: How the Ratios Behave

As your angle θ gets bigger (approaching 90°):
- The value of sin(θ) increases (from just over 0 towards 1).
- The value of cos(θ) decreases (from almost 1 towards 0).
- The value of tan(θ) increases (from just over 0 and grows very large).

Quick Review Box

Key Identities to Remember:
- $$tan(θ) = \frac{sin(θ)}{cos(θ)}$$
- $$sin^2(θ) + cos^2(θ) = 1$$
- $$sin(90° - θ) = cos(θ)$$

Key Takeaway

The trig ratios aren't random; they are connected by powerful and predictable rules. Knowing these identities can help you solve more complex problems.


Section 5: Trigonometry in the Real World!

Okay, let's use all this knowledge to solve some practical problems. This is where trigonometry really shines!

Angle of Elevation and Depression

These are special names for angles when you're looking up or down at something.

- Angle of Elevation: This is the angle you look UP from the horizontal line. (Imagine looking up from the ground to the top of a flagpole).
- Angle of Depression: This is the angle you look DOWN from the horizontal line. (Imagine standing on a cliff and looking down at a boat in the water).

Important Tip: The angle of elevation from the ground to an object is always equal to the angle of depression from the object back to the ground! They form a 'Z' shape with the parallel horizontal lines.

Gradients

A gradient measures the steepness of a slope, like a hill or a ramp. You might know it as "rise over run".
$$Gradient = \frac{Vertical Rise}{Horizontal Run}$$ Guess what? In a right-angled triangle, Rise is the Opposite side and Run is the Adjacent side. That means...
$$Gradient = \frac{Opposite}{Adjacent} = tan(θ)$$ So, the gradient of a slope is simply the tangent of its angle of inclination (θ)!

Bearings

Bearings are used in navigation to describe a direction. There are two main types you need to know.

1. True Bearings
- Measured clockwise from North (which is 000°). - Always written with three digits. - Example: East is 090°, South is 180°, and a direction of 45° would be written as 045°.

2. Conventional Bearings (Compass Bearings)
- Starts from either North (N) or South (S). - Then states an angle towards East (E) or West (W). - Example: N40°E means start by facing North, then turn 40 degrees towards the East. S20°W means start by facing South, then turn 20 degrees towards the West.

Key Takeaway

Trigonometry is a powerful tool for solving real-world problems. By setting up a right-angled triangle, you can find heights (using angles of elevation), steepness (using gradients), and directions (using bearings).