Foundations of Right Triangle Trigonometry

TL;DR

Right triangle trigonometry helps you find missing side lengths or angles in right-angled triangles using special ratios. These ratios (sine, cosine, and tangent) compare the lengths of a triangle's sides relative to a specific acute angle. Remember SOH CAH TOA to easily recall which sides go with which trigonometric function.

1. The Mental Model

Imagine you have a right-angled triangle, and you know some parts but not others. Trigonometry gives you a powerful toolkit to figure out those missing pieces – like an angle-measuring detective. It's all about how the sides of the triangle relate to its angles.

2. The Core Material

When you're working with right triangles, there are three main trigonometric ratios you'll use: sine (sin), cosine (cos), and tangent (tan). These ratios depend on an acute angle within the triangle and the lengths of the sides relative to that angle.

First, let's label the sides of a right triangle with respect to one of its acute angles:
* Hypotenuse: Always the longest side, opposite the right angle.
* Opposite: The side directly across from your chosen acute angle.
* Adjacent: The side next to your chosen acute angle that is not the hypotenuse.

Here's a diagram to help you visualize these relationships:

graph TD
    A["Right Angle (90°)"] --> B["Angle θ"]
    B --> C["Opposite Side"]
    A --> C
    A --> D["Adjacent Side"]
    B --> D
    C --- D
    B --- E["Hypotenuse"]
    A --- E
    style E fill:#fff,stroke:#333,stroke-width:2px,color:red;
    style C fill:#fff,stroke:#333,stroke-width:2px,color:blue;
    style D fill:#fff,stroke:#333,stroke-width:2px,color:green;
    linkStyle 0 stroke-width:0px;
    linkStyle 1 stroke-width:0px;
    linkStyle 2 stroke-width:0px;
    linkStyle 3 stroke-width:0px;
    linkStyle 4 stroke-width:0px;
    linkStyle 5 stroke-width:0px;
    linkStyle 6 stroke-width:0px;
    linkStyle 7 stroke-width:0px;
    subgraph "Right Triangle Sides"
        C -- "is opposite" --> B
        D -- "is adjacent to" --> B
        E -- "is hypotenuse of" --> B
    end

The SOH CAH TOA mnemonic

Photo by Tomás Asurmendi on Pexels

To remember the ratios, use the mnemonic SOH CAH TOA:

• SOH: Sin($\theta$) = Opposite / Hypotenuse
• CAH: Cos($\theta$) = Adjacent / Hypotenuse
• TOA: Tan($\theta$) = Opposite / Adjacent

These ratios are constant for a given angle, regardless of the size of the right triangle.

How to use them

Photo by Airam Dato-on on Pexels

• Finding a missing side: If you know one angle (other than the right angle) and one side length, you can use sin, cos, or tan to find another side.
• Finding a missing angle: If you know two side lengths, you can use the inverse functions (arcsin, arccos, arctan, often written as $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) to find one of the acute angles.

3. Worked Example

Let's say you have a right triangle where one acute angle ($\theta$) is 30 degrees, and the hypotenuse is 10 units long. You want to find the length of the side opposite the 30-degree angle.

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

   • Angle ($\theta$) = 30°
   • Hypotenuse = 10
   • Want to find: Opposite side

2. Choose the correct ratio:

   • You know the Hypotenuse and want to find the Opposite.
   • SOH (Sine = Opposite / Hypotenuse) is the right choice.

3. Set up the equation:

   • sin(30°) = Opposite / 10

4. Solve for the unknown:

   • Opposite = 10 * sin(30°)
   • Using a calculator, sin(30°) = 0.5
   • Opposite = 10 * 0.5
   • Opposite = 5 units

So, the side opposite the 30-degree angle is 5 units long.

4. Key Takeaways

• Trigonometry for right triangles involves sine, cosine, and tangent ratios.
• These ratios relate the lengths of sides to the acute angles in a right triangle.
• SOH CAH TOA is your best friend for remembering the ratios: Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj.
• Always correctly identify the Hypotenuse, Opposite, and Adjacent sides relative to the specific angle you're working with.
• Use the primary ratios (sin, cos, tan) to find side lengths when an angle and a side are known.
• Use inverse ratios ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) to find angles when two side lengths are known.

Common mistakes to avoid:
- Mixing up Opposite and Adjacent: These change depending on which acute angle you're referencing.
- Forgetting the Hypotenuse is always opposite the right angle: It's the longest side.
- Using the wrong ratio: Double-check SOH CAH TOA before setting up your equation.
- Calculator in wrong mode: Ensure your calculator is in "degrees" mode if your angles are in degrees, or "radians" mode if they're in radians.

5. Now Try It

You have a right triangle where the adjacent side to a 45-degree angle is 7 units long. Find the length of the opposite side. Work it out, and then double-check your answer by using the tangent ratio and finding that the opposite side should also be 7 units (because tan(45°) = 1).