9.1 Right-Angled Triangles
Understanding Pythagoras' Theorem
Pythagoras' Theorem is one of the most famous mathematical formulas. It relates the three sides of a right-angled triangle.
Pythagoras' Theorem is one of the most famous mathematical formulas. It relates the three sides of a right-angled triangle.
⚡ The Theorem:
Pythagoras' Theorem
a² + b² = c²
Where c is the hypotenuse (longest side opposite the right angle)
Key Terms:
• Right angle: 90° angle
• Hypotenuse: Longest side, opposite the right angle
• The theorem only works for right-angled triangles
• Right angle: 90° angle
• Hypotenuse: Longest side, opposite the right angle
• The theorem only works for right-angled triangles
Example 1: Finding the Hypotenuse
A right-angled triangle has sides of 3 cm and 4 cm. Find the length of the hypotenuse.
Formula:
$$a^2 + b^2 = c^2$$
Solution:
$3^2 + 4^2 = c^2$
$9 + 16 = c^2$
$25 = c^2$
$c = \sqrt{25} = 5$ cm
Answer: The hypotenuse is 5 cm
$$a^2 + b^2 = c^2$$
Solution:
$3^2 + 4^2 = c^2$
$9 + 16 = c^2$
$25 = c^2$
$c = \sqrt{25} = 5$ cm
Answer: The hypotenuse is 5 cm
Example 2: Finding a Shorter Side
A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.
Formula (rearranged):
$$a^2 = c^2 - b^2$$
Solution:
$a^2 = 13^2 - 5^2$
$a^2 = 169 - 25$
$a^2 = 144$
$a = \sqrt{144} = 12$ cm
Answer: The other side is 12 cm
Formula (rearranged):
$$a^2 = c^2 - b^2$$
Solution:
$a^2 = 13^2 - 5^2$
$a^2 = 169 - 25$
$a^2 = 144$
$a = \sqrt{144} = 12$ cm
Answer: The other side is 12 cm
Example 3: Problem Solving
A ladder 10 m long leans against a wall. The base of the ladder is 6 m from the wall. How high up the wall does the ladder reach?
Solution:
Using $a^2 + b^2 = c^2$:
$h^2 + 6^2 = 10^2$
$h^2 + 36 = 100$
$h^2 = 64$
$h = 8$ m
Answer: The ladder reaches 8 m up the wall
Using $a^2 + b^2 = c^2$:
$h^2 + 6^2 = 10^2$
$h^2 + 36 = 100$
$h^2 = 64$
$h = 8$ m
Answer: The ladder reaches 8 m up the wall
🎯 Pythagoras Practice
cm
cm
Understanding Trigonometry
Trigonometry helps us find angles and sides in right-angled triangles using ratios. The key is knowing which sides are which relative to the angle we're working with.
Trigonometry helps us find angles and sides in right-angled triangles using ratios. The key is knowing which sides are which relative to the angle we're working with.
⚡ SOH CAH TOA - The Three Ratios:
Sine (sin)
SOH
sin θ = Opposite / Hypotenuse
Cosine (cos)
CAH
cos θ = Adjacent / Hypotenuse
Tangent (tan)
TOA
tan θ = Opposite / Adjacent
Identifying the Sides:
• Hypotenuse (HYP): Always opposite the right angle (longest side)
• Opposite (OPP): The side opposite the angle you're working with
• Adjacent (ADJ): The side next to the angle but not the hypotenuse
• Hypotenuse (HYP): Always opposite the right angle (longest side)
• Opposite (OPP): The side opposite the angle you're working with
• Adjacent (ADJ): The side next to the angle but not the hypotenuse
Example 1: Finding a Side (using sin)
A right-angled triangle has hypotenuse 10 cm and angle 30°. Find the opposite side.
Step 1: Identify what we have
We know: angle = 30°, hypotenuse = 10 cm, want: opposite
Step 2: Choose the right ratio
sin involves opposite and hypotenuse → Use sin!
Step 3: Set up equation
$\sin(30°) = \frac{\text{opposite}}{10}$
Step 4: Solve
$\text{opposite} = 10 \times \sin(30°)$
$\text{opposite} = 10 \times 0.5 = 5$ cm
Answer: The opposite side is 5 cm
We know: angle = 30°, hypotenuse = 10 cm, want: opposite
Step 2: Choose the right ratio
sin involves opposite and hypotenuse → Use sin!
Step 3: Set up equation
$\sin(30°) = \frac{\text{opposite}}{10}$
Step 4: Solve
$\text{opposite} = 10 \times \sin(30°)$
$\text{opposite} = 10 \times 0.5 = 5$ cm
Answer: The opposite side is 5 cm
Example 2: Finding an Angle
A right-angled triangle has opposite side 8 cm and adjacent side 6 cm. Find the angle.
Step 1: Identify what we have
We know: opposite = 8 cm, adjacent = 6 cm
Step 2: Choose the right ratio
tan involves opposite and adjacent → Use tan!
Step 3: Set up equation
$\tan(\theta) = \frac{8}{6} = 1.333...$
Step 4: Use inverse tan (tan⁻¹)
$\theta = \tan^{-1}(1.333) = 53.1°$
Answer: The angle is 53.1°
Step 1: Identify what we have
We know: opposite = 8 cm, adjacent = 6 cm
Step 2: Choose the right ratio
tan involves opposite and adjacent → Use tan!
Step 3: Set up equation
$\tan(\theta) = \frac{8}{6} = 1.333...$
Step 4: Use inverse tan (tan⁻¹)
$\theta = \tan^{-1}(1.333) = 53.1°$
Answer: The angle is 53.1°
Example 3: Real-Life Problem
A 20 m tall building casts a shadow 15 m long. What is the angle of elevation of the sun?
Solution:
We have opposite (height) = 20 m and adjacent (shadow) = 15 m
$\tan(\theta) = \frac{20}{15} = 1.333$
$\theta = \tan^{-1}(1.333) = 53.1°$
Answer: The angle of elevation is 53.1°
Solution:
We have opposite (height) = 20 m and adjacent (shadow) = 15 m
$\tan(\theta) = \frac{20}{15} = 1.333$
$\theta = \tan^{-1}(1.333) = 53.1°$
Answer: The angle of elevation is 53.1°
💡 Trigonometry Tips:
• Always label the triangle first: Mark HYP, OPP, ADJ
• Finding an angle: Use inverse (sin⁻¹, cos⁻¹, tan⁻¹)
• SOH? CAH? TOA? Write down what you have and need, then pick
• Finding an angle: Use inverse (sin⁻¹, cos⁻¹, tan⁻¹)
• SOH? CAH? TOA? Write down what you have and need, then pick
🎯 Trigonometry Practice
Exact Trigonometric Values
For certain special angles (30°, 45°, 60°), we can write exact values without using a calculator.
For certain special angles (30°, 45°, 60°), we can write exact values without using a calculator.
⚡ Key Angles to Remember:
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | It's a secret |
Example 1: Using Exact Values
Calculate the exact value of: $10 \times \sin(30°)$
Solution:
From the table: $\sin(30°) = \frac{1}{2}$
$10 \times \sin(30°) = 10 \times \frac{1}{2} = 5$
Answer: 5 (exact answer)
Solution:
From the table: $\sin(30°) = \frac{1}{2}$
$10 \times \sin(30°) = 10 \times \frac{1}{2} = 5$
Answer: 5 (exact answer)
Example 2: 45-45-90 Triangle
An isosceles right-angled triangle has legs of length 1. Find the hypotenuse exactly.
Using Pythagoras:
$c^2 = 1^2 + 1^2 = 2$
$c = \sqrt{2}$
Answer: $\sqrt{2}$ (exact answer)
$c^2 = 1^2 + 1^2 = 2$
$c = \sqrt{2}$
Answer: $\sqrt{2}$ (exact answer)
Example 3: 30-60-90 Triangle
A right-angled triangle has a 30° angle, hypotenuse 10, and you need the opposite side.
Solution:
$\sin(30°) = \frac{\text{opposite}}{10}$
We know $\sin(30°) = \frac{1}{2}$
$\frac{1}{2} = \frac{\text{opposite}}{10}$
$\text{opposite} = 5$
Answer: 5 (exact answer)
Solution:
$\sin(30°) = \frac{\text{opposite}}{10}$
We know $\sin(30°) = \frac{1}{2}$
$\frac{1}{2} = \frac{\text{opposite}}{10}$
$\text{opposite} = 5$
Answer: 5 (exact answer)
💡 Memory Tricks:
• sin values increase: 0 → ½ → 1/√2 → √3/2 → 1
• cos values decrease: 1 → √3/2 → 1/√2 → ½ → 0
• 45° special: sin(45°) = cos(45°) = 1/√2
• 30° and 60°: sin(30°) = cos(60°) = ½
• tan(45°) = 1: Because opposite = adjacent
• cos values decrease: 1 → √3/2 → 1/√2 → ½ → 0
• 45° special: sin(45°) = cos(45°) = 1/√2
• 30° and 60°: sin(30°) = cos(60°) = ½
• tan(45°) = 1: Because opposite = adjacent
🎯 Exact Values
Real Life Uses:
• Architecture: Calculating roof angles and support beams
• Navigation: Finding distances and bearings
• Engineering: Designing ramps and slopes
• Surveying: Measuring heights of buildings and mountains
• Physics: Analyzing forces and motion on inclines
• Computer Graphics: 3D rotation and transformations
• Architecture: Calculating roof angles and support beams
• Navigation: Finding distances and bearings
• Engineering: Designing ramps and slopes
• Surveying: Measuring heights of buildings and mountains
• Physics: Analyzing forces and motion on inclines
• Computer Graphics: 3D rotation and transformations