Applications of Trigonometry: Your Guide to Solving Real-World Puzzles!
Hello! Ever wondered how people measure the height of a giant skyscraper without a super long measuring tape? Or how a ship navigates the vast ocean? The secret is trigonometry!
In this chapter, we're going to take your knowledge of sine, cosine, and tangent out of the textbook and into the real world. You'll learn how to find heights, distances, and directions using the power of triangles. It's like having a mathematical superpower to measure things you can't even reach. Let's get started!
A Super Quick Review: Remember SOH CAH TOA?
Before we dive into the cool applications, let's quickly refresh our memory on the basics. Everything we do here will be based on right-angled triangles.
Labelling the Sides
In any right-angled triangle, for a given angle (let's call it θ):
- The Hypotenuse (H) is always the longest side, opposite the right angle.
- The Opposite (O) side is the one directly across from the angle θ.
- The Adjacent (A) side is the one next to the angle θ (that isn't the hypotenuse).
The Magic Words: SOH CAH TOA
This is the best mnemonic to remember the three main trigonometric ratios:
SOH: Sine = Opposite / Hypotenuse $$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
CAH: Cosine = Adjacent / Hypotenuse $$ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$
TOA: Tangent = Opposite / Adjacent $$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $$
Key Takeaway
SOH CAH TOA is your best friend in trigonometry. It tells you which formula to use based on the sides you know and the side you want to find.
Solving Problems with Plane Figures (2D Shapes)
This is where we start using trigonometry to find missing lengths and angles in everyday situations. Don't worry if this seems tricky at first, we can solve almost any problem using a simple step-by-step process.
Your 5-Step Guide to Success:
- Draw a Diagram: Turn the word problem into a simple right-angled triangle.
- Label the Sides: Identify the Hypotenuse, Opposite, and Adjacent sides based on the angle in the problem.
- Choose Your Ratio: Look at what you know and what you need to find. Use SOH CAH TOA to pick the right formula.
- Write the Equation: Fill in the formula with the numbers from your problem.
- Solve It: Use your calculator to find the answer!
Example 1: Finding an Unknown Side
A 10-metre ladder leans against a wall, making an angle of 60° with the ground. How high up the wall does the ladder reach?
1. Draw: A triangle with the ladder as the hypotenuse.
2. Label:
- The angle is 60°.
- The ladder is the Hypotenuse (H) = 10 m.
- The height on the wall is Opposite (O) the angle = ? (let's call it h).
3. Choose: We have O and H, so we use SOH (Sine).
4. Write: $$ \sin(60^\circ) = \frac{h}{10} $$
5. Solve: $$ h = 10 \times \sin(60^\circ) $$
$$ h \approx 8.66 \text{ m} $$
So, the ladder reaches about 8.66 metres up the wall!
Example 2: Finding an Unknown Angle
You are standing 20 metres away from the base of a tree. The height of the tree is 30 metres. What is the angle you have to look up to see the top of the tree?
1. Draw: A triangle with you, the tree base, and the treetop.
2. Label:
- The height of the tree is the Opposite (O) side = 30 m.
- The distance from the tree is the Adjacent (A) side = 20 m.
- We need to find the angle θ.
3. Choose: We have O and A, so we use TOA (Tangent).
4. Write: $$ \tan(\theta) = \frac{30}{20} = 1.5 $$
5. Solve: To find the angle, we use the inverse tangent button on our calculator (tan⁻¹).
$$ \theta = \tan^{-1}(1.5) $$
$$ \theta \approx 56.3^\circ $$
So, you have to look up at an angle of about 56.3 degrees.
Key Takeaway
To solve any right-angled triangle problem, just follow the 5 steps: Draw, Label, Choose, Write, Solve!
Looking Up and Down: Elevation & Depression
These are special names for angles we use when looking up or down at objects. They are super useful for figuring out heights and distances.
Angle of Elevation
This is the angle you look UP from a horizontal line.
Memory Aid: Think of an 'Elevator' which goes UP.
Example: Imagine you are standing on the ground looking up at the top of a building. The angle your eyes make with the flat ground is the angle of elevation.
Angle of Depression
This is the angle you look DOWN from a horizontal line.
Memory Aid: Feeling 'Depressed' can make you feel 'Down'.
Example: You are standing on top of a cliff looking down at a boat in the sea. The angle from the horizontal line of your eyesight down to the boat is the angle of depression.
Common Mistake to Avoid!
The angle of depression is measured from the horizontal line outside the triangle. BUT, because of geometry (remember alternate 'Z' angles?), the angle of depression from the top is always equal to the angle of elevation from the bottom. This is a very helpful trick!
Key Takeaway
Elevation is looking UP from the horizontal. Depression is looking DOWN from the horizontal. They are both measured from a flat, horizontal line.
How Steep Is It? Gradients
A gradient is simply a number that tells us how steep something is, like a hill or a ramp. It's often described as "rise over run".
$$ \text{Gradient} = \frac{\text{Vertical Change (Rise)}}{\text{Horizontal Change (Run)}} $$
The Connection to Trigonometry
Look at the formula for gradient. In a right-angled triangle, the "Rise" is the Opposite side and the "Run" is the Adjacent side. Which trig ratio is Opposite/Adjacent? It's Tangent!
This gives us a super important formula:
$$ \text{Gradient} = \tan(\theta) $$
Here, θ is called the angle of inclination. It's the angle the slope makes with the horizontal.
Example:
A wheelchair ramp has a gradient of 1/12. What is its angle of inclination?
We know: $$ \tan(\theta) = \text{Gradient} $$
$$ \tan(\theta) = \frac{1}{12} $$
$$ \theta = \tan^{-1}\left(\frac{1}{12}\right) $$
$$ \theta \approx 4.76^\circ $$
The ramp has a gentle slope of about 4.76 degrees.
Did you know?
Engineers use gradients all the time to design roads that are safe for cars and roofs that allow rain to drain off properly.
Key Takeaway
Gradient measures steepness and is equal to the tangent of the angle of inclination. If you know one, you can always find the other!
Finding Your Way: Bearings
Bearings are used in navigation by pilots and sailors to describe a precise direction. We will learn two types.
Type 1: Three-Figure Bearings (or True Bearings)
This type has three simple rules:
- Always measure from North.
- Always measure in a clockwise direction.
- Always write your answer using three figures. (e.g., 50° is written as 050°).
Example: A bearing of 135° means you start facing North, then turn 135° clockwise.
Type 2: Compass Bearings
This type uses North/South and East/West directions.
- You start by facing either North (N) or South (S).
- Then, you turn a certain angle towards either East (E) or West (W).
Example: The bearing N40°E means "Start at North, then turn 40° towards the East".
Example: The bearing S20°W means "Start at South, then turn 20° towards the West".
Example Problem:
A ship sails 15 km from a port on a bearing of 060°. How far north of the port is the ship?
1. Draw: Draw a North line. Measure 60° clockwise. Draw a line 15 km long. Make a right-angled triangle by drawing a line back to the North-South line.
2. Label: The angle inside our triangle at the port is 60°. The hypotenuse is 15 km. The "how far north" part is the Adjacent (A) side.
3. Choose: We have A and H, so we use CAH (Cosine).
4. Write: $$ \cos(60^\circ) = \frac{\text{Adjacent}}{15} $$
5. Solve: $$ \text{Adjacent} = 15 \times \cos(60^\circ) = 7.5 \text{ km} $$
The ship is 7.5 km north of the port.
Key Takeaway
Bearings tell us direction. Remember the rules:
- True Bearings: From North, Clockwise, 3 Figures.
- Compass Bearings: From N or S, turn towards E or W.