Introduction to Right-Angled Triangles and Basic Trigonometric Ratios

TL;DR

Right-angled triangles have one 90-degree angle, and their sides have special names. Trigonometric ratios (sine, cosine, tangent) describe the relationships between the angles and the side lengths. These ratios are fundamental for solving problems involving unknown angles or side lengths in right triangles.

1. The Mental Model

Imagine a ramp or a ladder leaning against a wall; it forms a right-angled triangle. We're going to learn how to relate the angles of that ramp or ladder to its length and how high it reaches.

2. The Core Material

You're probably familiar with triangles, but a right-angled triangle is special because one of its angles is exactly 90 degrees (a right angle). This right angle is super important because it helps us define the other parts of the triangle.

Identifying the Sides

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In a right-angled triangle, the sides have specific names relative to a chosen non-right angle (we usually call this angle $\theta$ "theta").

• Hypotenuse: This is always the longest side and is always opposite the 90-degree angle.
• Opposite: This side is directly across from your chosen angle $\theta$.
• Adjacent: This side is next to your chosen angle $\theta$ but isn't the hypotenuse.

It's crucial to remember that the "opposite" and "adjacent" sides change if you choose a different non-right angle in the same triangle. The hypotenuse always stays the same.

graph TD
    A["Right-Angled Triangle"] --> B["Identify 90° Angle"]
    B --> C["Hypotenuse (Opposite 90°)"]
    B --> D{"Choose Reference Angle θ"}
    D --> E["Opposite Side (Opposite θ)"]
    D --> F["Adjacent Side (Next to θ, not Hypotenuse)"]
    C & E & F --> G["Define Trig Ratios (SOH CAH TOA)"]

The Basic Trigonometric Ratios (SOH CAH TOA)

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Once you've identified the sides, you can define the three basic trigonometric ratios: Sine (sin), Cosine (cos), and Tangent (tan). A common mnemonic to remember them is SOH CAH TOA:

• 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

These ratios give you a way to find an unknown side length if you know an angle and one side, or to find an unknown angle if you know two side lengths. Your calculator will have sin, cos, and tan buttons (and their inverses: sin^-1, cos^-1, tan^-1 which you use to find angles). Make sure your calculator is in "degree" mode for these problems unless otherwise specified.

3. Worked Example

Let's say you have a right-angled triangle. The angle $\theta$ is 30 degrees. 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:

   • Angle $\theta = 30^\circ$
   • Hypotenuse = 10
   • You want to find the Opposite side.

2. Choose the correct ratio: You know the Hypotenuse and want to find the Opposite. The ratio that connects Opposite and Hypotenuse is Sine (SOH).

   • sin(angle) = opposite / hypotenuse

3. Plug in the values:

   • sin(30°) = opposite / 10

4. Solve for the unknown:

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

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

4. Key Takeaways

• A right-angled triangle always has one 90-degree angle.
• The hypotenuse is always the longest side, opposite the 90-degree angle.
• "Opposite" and "adjacent" sides are defined relative to a chosen non-right angle.
• SOH CAH TOA helps you remember the three basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
• You use these ratios to find unknown side lengths or angles in right-angled triangles.

Common Mistakes to Avoid:

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• Mixing up the opposite and adjacent sides: always define them relative to the chosen angle.
• Forgetting that the hypotenuse is always opposite the 90-degree angle.
• Using the trigonometric ratios with the 90-degree angle itself (they only apply to the other two acute angles).
• Calculator mode: ensure your calculator is in "DEG" (degree) mode, not "RAD" (radian) mode, for typical problems.

5. Now Try It

Draw a right-angled triangle. Label one of the non-right angles as 60 degrees. Label the side adjacent to the 60-degree angle as 7 units long. Now, using what you've learned, calculate the length of the hypotenuse.

What success looks like: You should be able to identify which trigonometric ratio to use (hint: CAH), set up the equation, and correctly calculate the hypotenuse's length. The answer should be approximately 14 units.