Hello Future Geometry Expert!
Welcome to the chapter on Trigonometry and Pythagoras’ Theorem! Don't worry if these words look intimidating—they are just fancy names for some of the most powerful tools in geometry. We are going to learn how to measure distances and angles without ever having to touch a ruler or protractor!
These skills are essential because they allow engineers, architects, and even video game designers to calculate height, slopes, and distances accurately. Let’s dive in!
1. Pythagoras' Theorem: The Safety Net for Right Triangles
1.1 What is Pythagoras' Theorem?
This theorem only works for right-angled triangles (triangles containing a 90° angle). It describes the relationship between the lengths of the three sides.
Key Concept: If you square the two shorter sides (legs) and add them together, the result will equal the square of the longest side (the hypotenuse).
- The Hypotenuse (c): Always the longest side, and always opposite the right angle.
- The Other two sides (a and b): These are the legs that meet at the right angle.
The Formula:
\[ a^2 + b^2 = c^2 \]
1.2 Finding the Hypotenuse (c)
When you know the lengths of the two shorter sides, you use the formula directly.
Step-by-Step: Finding the Hypotenuse
- Square the length of side a.
- Square the length of side b.
- Add the results from Step 1 and 2.
- Take the square root of the total to find c.
Example: If a = 3 cm and b = 4 cm, what is c?
\[ 3^2 + 4^2 = c^2 \]
\[ 9 + 16 = c^2 \]
\[ 25 = c^2 \]
\[ c = \sqrt{25} = 5 \text{ cm} \]
1.3 Finding a Shorter Side (a or b)
If you know the hypotenuse and one other side, you need to rearrange the formula.
Rearranged Formulas:
\[ a^2 = c^2 - b^2 \]
\[ b^2 = c^2 - a^2 \]
Remember: To find a shorter side, you must subtract the known short side squared from the hypotenuse squared.
Step-by-Step: Finding a Shorter Side
- Square the length of the Hypotenuse (c).
- Square the length of the known short side (a or b).
- Subtract the smaller squared number from the larger squared number.
- Take the square root of the result.
! Common Mistake Alert !
Never add the squares when looking for a shorter side! The hypotenuse (c) must always be the longest side, so your answer must be smaller than c.
1.4 Application in 3D Problems (IGCSE Extension)
Pythagoras' theorem can be used repeatedly to find the length of a diagonal line inside a 3D shape (like a cuboid).
To find the space diagonal (the longest line from one corner to the opposite far corner), you first calculate the diagonal across the base (2D), and then use that answer, along with the height, in a second application of the theorem.
Key Takeaway for Pythagoras: \(a^2 + b^2 = c^2\). Use addition (+) to find the hypotenuse, and subtraction (-) to find a shorter leg. It only works for right-angled triangles.
2. Trigonometry: Measuring Angles and Sides
What if you need to find an angle, or you know an angle but need to find a side? Pythagoras can't help! This is where Trigonometry comes in. It connects the angles of a right triangle with the ratios of its side lengths.
2.1 Naming the Sides (The Crucial First Step!)
When using trigonometry, the names of the sides change depending on which angle (other than the 90° angle) you are focusing on.
1. Hypotenuse (H):
Always the longest side, opposite the 90° angle.
2. Opposite (O):
The side across from the angle (\(\theta\)) you are working with.
3. Adjacent (A):
The side next to the angle (\(\theta\)) that is not the hypotenuse.
Imagine standing at the angle \(\theta\). The side you are looking directly at is the Opposite. The ground right next to your feet is the Adjacent.
2.2 The Three Trigonometric Ratios (SOH CAH TOA)
The three main ratios are Sine, Cosine, and Tangent. They define the ratio of two sides relative to a specific angle.
The Mnemonic: SOH CAH TOA
This is the most important memory tool you will learn today!
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
Did You Know? The word "Sine" comes from a Latin word meaning 'bay' or 'curve', reflecting the ancient use of these relationships in astronomy.
Quick Review Box:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
3. Using Trigonometry to Find Unknown Sides
If you know one angle (\(\theta\)) and the length of one side, you can find the length of any other side.
Step-by-Step: Finding a Side Length
- Identify: Circle the known angle \(\theta\).
- Label: Label the three sides (O, A, H) relative to \(\theta\).
- Choose the Ratio: Look at the side you know and the side you want to find. Use SOH CAH TOA to select the ratio that includes both of those letters.
- Substitute: Put the values into the chosen formula.
- Solve: Use simple algebra to isolate the unknown side (x).
Example 1: Finding the Opposite Side (Using Sine)
We know the angle \(\theta = 30^\circ\), and the Hypotenuse (H) = 10. We want to find the Opposite (O).
1. We have O and H. We use SOH (Sine).
\[ \sin(\theta) = \frac{O}{H} \]
\[ \sin(30^\circ) = \frac{x}{10} \]
2. To solve for x, multiply both sides by 10:
\[ x = 10 \times \sin(30^\circ) \]
\[ x = 10 \times 0.5 = 5 \]
Algebra Trick:
If the unknown side (x) is on the top of the fraction, you multiply the trig function by the side length.
If the unknown side (x) is on the bottom of the fraction, you divide the known side length by the trig function.
! Precision Tip !
Always use the full calculator value for your trigonometric functions (e.g., \(\cos(25^\circ)\)) and only round your final answer to the required degree of accuracy (usually 3 significant figures).
Key Takeaway for Finding Sides: Label O, A, H. Choose the ratio that uses the two sides involved (one known, one unknown). Solve algebraically.
4. Using Trigonometry to Find Unknown Angles
If you know the lengths of at least two sides of the right triangle, you can find any acute angle.
4.1 Introducing Inverse Functions
When you want to find an angle, you use the inverse functions, sometimes called arc functions. These are the buttons on your calculator marked \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) (usually accessed by pressing 'Shift' or '2nd' before the trig button).
Analogy: If \(\sin(30^\circ) = 0.5\), then \(\sin^{-1}(0.5) = 30^\circ\). The inverse function is like the "undo" button for the ratio!
Step-by-Step: Finding an Angle
- Identify: Label the known sides (O, A, H) relative to the angle \(\theta\) you want to find.
- Choose the Ratio: Select the ratio (SOH CAH TOA) that uses your two known sides.
- Calculate the Ratio: Substitute the side lengths and calculate the decimal ratio.
- Apply Inverse: Use the appropriate inverse function (\(\sin^{-1}, \cos^{-1}, \tan^{-1}\)) on the ratio value to find the angle \(\theta\).
Example 2: Finding an Angle (Using Tangent)
The Opposite side (O) = 8 cm and the Adjacent side (A) = 5 cm. Find angle \(\theta\).
1. We have O and A. We use TOA (Tangent).
\[ \tan(\theta) = \frac{O}{A} \]
\[ \tan(\theta) = \frac{8}{5} \]
2. Calculate the ratio: \(8 \div 5 = 1.6\)
3. Apply the inverse tangent function:
\[ \theta = \tan^{-1}(1.6) \]
\[ \theta \approx 57.994...^\circ \]
4. Rounded to 3 significant figures: \(\theta = 58.0^\circ\)
Key Takeaway for Finding Angles: Label O, A, H. Set up the ratio. Use the inverse function (\(\sin^{-1}, \cos^{-1}, \tan^{-1}\)) to calculate the angle.
5. Angles of Elevation and Depression
Trigonometry is often used to solve "word problems" involving height and distance. Two key terms you must understand are related to a horizontal line of sight.
5.1 Angle of Elevation
The Angle of Elevation is the angle measured upwards from a horizontal line to a point above the observer.
Imagine looking up at a bird flying.
5.2 Angle of Depression
The Angle of Depression is the angle measured downwards from a horizontal line to a point below the observer.
Imagine looking down from a cliff at a boat on the water.
Important Geometric Rule:
Because the horizontal line of sight (at the observer) and the ground (or base) are parallel, the Angle of Depression from the top is always equal to the Angle of Elevation from the bottom (alternate angles).
This means if a tower is 50m high, and the angle of depression from the top of the tower to a point on the ground is \(15^\circ\), then the angle of elevation from that point on the ground to the top of the tower is also \(15^\circ\).
! Troubleshooting Word Problems !
When solving word problems:
- Always draw a diagram! Label the 90° angle clearly.
- Place the given angle.
- Label the known and unknown sides (O, A, H).
- Use SOH CAH TOA to set up your equation.
Final Review: Right-Angled Geometry Tools
We now have two powerful tools for right-angled triangles:
- Pythagoras (\(a^2 + b^2 = c^2\)): Used when you know 2 side lengths and need to find the 3rd (no angles involved).
- Trigonometry (SOH CAH TOA): Used when you know 1 angle and 1 side, or when you know 2 sides and need to find an angle.
Congratulations! You have mastered the fundamentals of right-angled geometry. Keep practicing the steps, and remember the magic words: SOH CAH TOA!