Welcome to the World of Trigonometry!
Hey there! Get ready to dive into one of the most useful parts of mathematics: Trigonometry. The name might sound complicated, but don't worry! "Trigonometry" just means "triangle measurement".
In this chapter, you'll learn how to find missing sides and angles in triangles without even measuring them directly. It's like having a mathematical superpower! This is used in real life for things like building skyscrapers, designing video games, navigating ships, and even creating animated movies. So, let's get started!
Section 1: The Right-Angled Triangle - Our Superhero!
Trigonometry starts with a special kind of triangle: the right-angled triangle. This is any triangle that has one angle of exactly 90 degrees (a right angle).
Labelling the Sides
Before we can do anything, we need to give the sides of our triangle special names. The names change depending on which angle we're focusing on. Let's call our focus angle theta ($$\theta$$), which is just a Greek letter we often use for unknown angles.
Quick Review: The Three Sides
- Hypotenuse (H): This one is the easiest to spot. It's always the longest side, and it's always opposite the right angle. Think of it as the "boss" of the triangle.
- Opposite (O): This is the side directly across from our focus angle ($$\theta$$). If you were "standing" at angle $$\theta$$, this side would be opposite you.
- Adjacent (A): This word just means "next to". This is the side that is next to our focus angle ($$\theta$$), but is not the hypotenuse.
Example: Imagine you are standing at angle A. The side BC is 'opposite' you. The side AB is 'adjacent' to you. AC is always the 'hypotenuse'. If you walk over and stand at angle C, the labels change! Now, AB is 'opposite' and BC is 'adjacent'. The hypotenuse always stays the same!
Key Takeaway
The names Opposite and Adjacent depend on which angle you are looking from. The Hypotenuse is always the side opposite the right angle.
Section 2: The Three Musketeers - Sine, Cosine, and Tangent
Now for the main event! We have three key tools in trigonometry. They are called Sine (sin), Cosine (cos), and Tangent (tan). They might sound strange, but they are just names for the ratios of the side lengths we just learned about.
The Magic Mnemonic: SOH CAH TOA
This is the most important thing to remember in trigonometry. It's a silly-sounding phrase that tells you exactly what each ratio is. Say it out loud: "SOH - CAH - TOA".
- SOH stands for: Sine = Opposite / Hypotenuse
$$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
- CAH stands for: Cosine = Adjacent / Hypotenuse
$$ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$
- TOA stands for: Tangent = Opposite / Adjacent
$$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $$
Quick Review Box
SOH: $$ \sin(\theta) = \frac{O}{H} $$
CAH: $$ \cos(\theta) = \frac{A}{H} $$
TOA: $$ \tan(\theta) = \frac{O}{A} $$
Key Takeaway
Sine, Cosine and Tangent are just ratios of the lengths of the sides of a right-angled triangle. Remember SOH CAH TOA to get the formulas right every time!
Section 3: Using Your Calculator - Your New Best Friend
Your calculator knows all about SOH CAH TOA. It has buttons for `sin`, `cos`, and `tan`.
Finding a Ratio from an Angle
If you know an angle, you can find its ratio. For example, to find the sine of 40 degrees:
- Make sure your calculator is in DEGREE mode. You should see a "D" or "DEG" at the top of the screen. If you see "R" or "G", it's in the wrong mode!
- Press the `sin` button.
- Type in `40`.
- Press `=`. You should get something like 0.6427...
Finding an Angle from a Ratio
What if you know the ratio but need to find the angle? We use the "inverse" functions. Look for `sin⁻¹`, `cos⁻¹`, and `tan⁻¹` on your calculator. You usually have to press a `SHIFT` or `2nd` key first.
Example: You calculated that $$ \sin(\theta) = 0.5 $$. To find the angle $$ \theta $$:
- Press `SHIFT` and then the `sin` button to get `sin⁻¹`.
- Type in `0.5`.
- Press `=`. The calculator will tell you the angle is 30 degrees.
Common Mistake Alert!
Always, always, always check that your calculator is in DEG mode! If it's in RAD (Radians) or GRA (Gradians), you will get completely different answers, and they will be wrong for these problems.
Section 4: Solving Problems - Putting It All Together!
This is where the magic happens! We can use SOH CAH TOA to find missing sides and angles.
Part A: Finding a Missing Side
Let's find the length of side x in a triangle where the hypotenuse is 10 cm and an angle is 35°.
- Label the sides: From the 35° angle, x is the Opposite side, and 10 cm is the Hypotenuse.
- Choose your ratio: We have O and H. Look at SOH CAH TOA. The one with O and H is... SOH! So we use Sine.
- Write the equation:
$$ \sin(35^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{x}{10} $$ - Solve for x: To get x by itself, multiply both sides by 10.
$$ x = 10 \times \sin(35^\circ) $$
Use your calculator: $$ x \approx 5.74 $$ cm.
Part B: Finding a Missing Angle
Let's find the angle $$ \theta $$ in a triangle where the Opposite side is 5 cm and the Adjacent side is 8 cm.
- Label the sides: We know the Opposite (5) and the Adjacent (8).
- Choose your ratio: We have O and A. The one with O and A is... TOA! So we use Tangent.
- Write the equation:
$$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{8} $$ - Solve for $$ \theta $$: Calculate the ratio: $$ \frac{5}{8} = 0.625 $$.
Now use the inverse tangent function on your calculator:
$$ \theta = \tan^{-1}(0.625) $$
Your calculator will show: $$ \theta \approx 32^\circ $$.
Section 5: Special Angles (The VIPs of Trigonometry)
There are three angles that are so special in maths, we learn their trigonometric ratios by heart: 30°, 45°, and 60°. They give neat, exact answers instead of long decimals!
The Table of Special Values
| Angle ($$\theta$$) | $$ \sin(\theta) $$ | $$ \cos(\theta) $$ | $$ \tan(\theta) $$ | | :---: | :---: | :---: | :---: | | 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?
You can figure out all these values just by drawing two simple triangles! One is an isosceles right-angled triangle (with sides 1, 1, $$ \sqrt{2} $$) and the other is half an equilateral triangle (with sides 1, 2, $$ \sqrt{3} $$). Try using SOH CAH TOA on them to see where the values in the table come from!
Section 6: Properties and Relationships - The Secret Connections
The three ratios are related to each other in some cool ways. Don't worry if these seem tricky at first, they are shortcuts that become very useful later on!
The Quotient Identity
This connects all three ratios. It turns out that Tangent is just Sine divided by Cosine!
$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
The Pythagorean Identity
This one comes from Pythagoras' Theorem and is super powerful.
$$ \sin^2(\theta) + \cos^2(\theta) = 1 $$
(Note: $$ \sin^2(\theta) $$ is just a short way of writing $$ (\sin(\theta))^2 $$)
Complementary Angles
In a right-angled triangle, the two acute angles always add up to 90°. This creates a neat pattern:
- $$ \sin(\theta) = \cos(90^\circ - \theta) $$ (The sine of an angle is the cosine of its complement)
- $$ \cos(\theta) = \sin(90^\circ - \theta) $$ (The cosine of an angle is the sine of its complement)
- $$ \tan(\theta) = \frac{1}{\tan(90^\circ - \theta)} $$
For example, $$ \sin(30^\circ) $$ is exactly the same as $$ \cos(60^\circ) $$. Try it on your calculator!
Section 7: Real-World Trigonometry - It's Everywhere!
Let's see how trigonometry is used to solve real problems.
Angles of Elevation and Depression
These are special names for angles when you look up or down at something.
- Angle of Elevation: If you are looking up at an object, this is the angle between the horizontal ground and your line of sight. (Think "elevate" means to lift up).
- Angle of Depression: If you are on top of something looking down, this is the angle between the horizontal line and your line of sight. (Think "depressed" means feeling down).
Example: A person standing 20 metres from a tree measures the angle of elevation to the top of the tree as 40°. How tall is the tree?
You have the Adjacent side (20 m) and want to find the Opposite side (the height). Use TOA!
$$ \tan(40^\circ) = \frac{\text{height}}{20} $$
$$ \text{height} = 20 \times \tan(40^\circ) \approx 16.8 $$ metres.
Gradients
A gradient measures how steep a slope is. You may have seen it on road signs. The gradient is the "rise" divided by the "run". Guess what? That's the same as Opposite / Adjacent!
Gradient = $$ \tan(\theta) $$, where $$ \theta $$ is the angle of the slope.
Bearings
Bearings are used in navigation to describe a direction. There are two main types:
- True Bearings: An angle from 000° to 359°, measured clockwise from North. It must always be written with 3 figures. Example: 050° is 50° clockwise from North. 220° is 220° clockwise from North.
- Compass Bearings: We start at North (N) or South (S) and then turn a certain number of degrees towards East (E) or West (W). Example: N25°E means start by facing North, then turn 25° towards the East.
Key Takeaway
Trigonometry helps us connect angles and distances in the real world. By drawing a right-angled triangle, you can solve problems about height, distance, steepness, and direction. You've got this!