Foundations of Right-Angled Trigonometry

TL;DR

Right-angled trigonometry is all about relating angles and side lengths in triangles that have a 90-degree corner. We use special ratios called sine, cosine, and tangent (SOH CAH TOA) to find missing sides or angles. These fundamental tools are essential for solving many real-world problems involving distances and heights.

1. The Mental Model

Imagine you're standing still, looking at a tall building. If you know how far you are from the building and the angle you have to tilt your head to see the top, trigonometry lets you figure out the building's height without climbing it. It's like having a special ruler and protractor that work together.

2. The Core Material

Right-angled triangles are special because one of their angles is exactly 90 degrees. This fixed angle allows us to define consistent relationships between the other two angles and the lengths of the sides.

Naming the Sides

Photo by Matt Webster on Pexels

First, you need to correctly identify the sides relative to a specific non-90-degree angle you're interested in (let's call it $\theta$).

• Hypotenuse: This is always the longest side and is directly opposite the 90-degree angle. It never changes, no matter which angle $\theta$ you pick.
• Opposite: This side is directly across from your chosen angle $\theta$.
• Adjacent: This side is next to your chosen angle $\theta$ and is not the hypotenuse.

graph TD
    A["Choose an Angle (not 90°)"] --> B{"Identify Sides relative to that Angle"};
    B --> C["Side opposite 90°: **Hypotenuse** (longest)"];
    B --> D["Side opposite chosen angle: **Opposite**"];
    B --> E["Side next to chosen angle (not hypotenuse): **Adjacent**"];
    C & D & E --> F["Apply SOH CAH TOA"];

The Trigonometric Ratios (SOH CAH TOA)

Photo by Tomás Asurmendi on Pexels

These are the three fundamental ratios that connect an angle in a right triangle to the lengths of its sides. You'll often hear the mnemonic "SOH CAH TOA" to remember them.

• SOH: Sine($\theta$) = Opposite / Hypotenuse
  • sin(angle) = Opposite / Hypotenuse
• CAH: Cosine($\theta$) = Adjacent / Hypotenuse
  • cos(angle) = Adjacent / Hypotenuse
• TOA: Tangent($\theta$) = Opposite / Adjacent
  • tan(angle) = Opposite / Adjacent

You use these ratios in two main ways:
1. Finding a missing side: If you know an angle and one side, you can use the appropriate ratio to find another side.
2. Finding a missing angle: If you know two sides, you can use the inverse trigonometric functions (like arcsin, arccos, arctan or sin⁻¹, cos⁻¹, tan⁻¹) to find the angle.

Inverse Functions

Photo by https://kaboompics.com/ on Pexels

When you want to find an angle, you'll use the inverse trigonometric functions. Your calculator usually has buttons like sin⁻¹, cos⁻¹, and tan⁻¹.

• If sin(angle) = ratio, then angle = sin⁻¹(ratio)
• If cos(angle) = ratio, then angle = cos⁻¹(ratio)
• If tan(angle) = ratio, then angle = tan⁻¹(ratio)

Always make sure your calculator is in "degree" mode for these problems, not "radian" mode, unless specified.

3. Worked Example

Let's say you have a right-angled triangle.
* The angle is 30 degrees.
* The side opposite the 30-degree angle is 5 units long.
* You want to find the length of the hypotenuse.

1. Identify what you know and what you want to find:

   • Angle ($\theta$) = 30°
   • Opposite side = 5
   • Hypotenuse = ? (let's call it h)

2. Choose the correct ratio: You know the Opposite and want the Hypotenuse. The SOH part of SOH CAH TOA relates Opposite and Hypotenuse: Sine.

   • sin(angle) = Opposite / Hypotenuse

3. Plug in the values:

   • sin(30°) = 5 / h

4. Solve for h:

   • h = 5 / sin(30°)
   • You know that sin(30°) = 0.5 (or you'd use a calculator).
   • h = 5 / 0.5
   • h = 10

So, the hypotenuse is 10 units long.

4. Key Takeaways

• Always identify the hypotenuse first; it's opposite the 90-degree angle.
• The terms "opposite" and "adjacent" depend entirely on which non-90-degree angle you're focusing on.
• SOH CAH TOA is your mnemonic for remembering the three basic trigonometric ratios.
• Use sine for Opposite/Hypotenuse, cosine for Adjacent/Hypotenuse, and tangent for Opposite/Adjacent.
• Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) when you need to find an angle.
• Ensure your calculator is in "DEGREE" mode for these problems.

Common Mistakes to Avoid:
* Confusing the opposite and adjacent sides by not clearly picking your reference angle.
* Using the 90-degree angle with SOH CAH TOA; these ratios only apply to the other two acute angles.
* Incorrectly setting up the ratio (e.g., mixing up the numerator and denominator).
* Forgetting to use inverse functions when finding angles or regular functions when finding sides.

5. Now Try It

You're looking at a flagpole. You are 20 meters away from its base. The angle of elevation (the angle from the ground up to the top of the flagpole) is 45 degrees.

1. Draw a right-angled triangle representing this situation. Label the known angle and side.
2. Identify which side is opposite, adjacent, and the hypotenuse relative to the 45-degree angle.
3. Choose the correct trigonometric ratio to find the height of the flagpole (the side opposite the 45-degree angle).
4. Calculate the height of the flagpole.

What success looks like: You should find that the height of the flagpole is 20 meters.