Foundational Concepts & Right-Angled Triangles

TL;DR

Trigonometry is all about relationships between angles and sides in triangles, especially right-angled triangles. The core functions – sine, cosine, and tangent – help you find unknown sides or angles using ratios. Remember SOH CAH TOA to apply these ratios correctly.

1. The Mental Model

Imagine you're trying to figure out the height of a building, but you can't measure it directly. If you know how far you are from it and the angle to its top, trigonometry gives you the tools to find that height. It's like having a special ruler for angles and distances.

2. The Core Material

Trigonometry primarily deals with triangles. While it applies to all triangles, it starts with the easiest one: the right-angled triangle. This is a triangle with one angle exactly 90 degrees.

Identifying Parts of a Right-Angled Triangle

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First, let's name the sides relative to one of the acute angles (the angles less than 90 degrees).

• Hypotenuse: Always the longest side, and it's always opposite the 90-degree angle.
• Opposite: The side directly across from the angle you're interested in.
• Adjacent: The side next to the angle you're interested in that isn't the hypotenuse.

graph TD
    A["Right-Angled Triangle Concepts"] --> B["Identify Sides"]
    B --> C["Hypotenuse (longest, opp 90°)"]
    B --> D["Opposite (opp selected angle)"]
    B --> E["Adjacent (next to selected angle, not hypotenuse)"]
    A --> F["Trigonometric Ratios (SOH CAH TOA)"]
    F --> G["Sine (Opposite / Hypotenuse)"]
    F --> H["Cosine (Adjacent / Hypotenuse)"]
    F --> I["Tangent (Opposite / Adjacent)"]
    F --> J["Solving Problems (find unknown side/angle)"]

The Three Basic Trigonometric Ratios: SOH CAH TOA

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These three ratios are the foundation of trigonometry. They relate the angles of a right triangle to the lengths of its sides.

1. Sine (sin): The ratio of the Opposite side to the Hypotenuse.
   • $\sin(\theta) = \text{Opposite} / \text{Hypotenuse}$
2. Cosine (cos): The ratio of the Adjacent side to the Hypotenuse.
   • $\cos(\theta) = \text{Adjacent} / \text{Hypotenuse}$
3. Tangent (tan): The ratio of the Opposite side to the Adjacent side.
   • $\tan(\theta) = \text{Opposite} / \text{Adjacent}$

A handy mnemonic to remember these is SOH CAH TOA:
* Sine = Opposite / Hypotenuse
* Cosine = Adjacent / Hypotenuse
* Tangent = Opposite / Adjacent

Using Trigonometric Ratios

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You can use these ratios to:
* Find an unknown side length if you know an angle and one side.
* Find an unknown angle if you know two side lengths.

To find an angle, you'll use the inverse trigonometric functions: $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$. These "undo" sine, cosine, and tangent, giving you the angle that corresponds to a particular ratio.

3. Worked Example

Let's say you have a right-angled triangle.
* One angle 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 and what you want to find:

   • Angle ($\theta$) = 30 degrees
   • Hypotenuse = 10
   • We want to find the Opposite side.

2. Choose the correct ratio:

   • We know the Hypotenuse and want to find the Opposite. SOH CAH TOA tells us Sine relates Opposite and Hypotenuse. So, we use sine.

3. Set up the equation:

   • $\sin(\theta) = \text{Opposite} / \text{Hypotenuse}$
   • $\sin(30^\circ) = \text{Opposite} / 10$

4. Solve for the unknown:

   • From a calculator (or by memory), $\sin(30^\circ) = 0.5$.
   • $0.5 = \text{Opposite} / 10$
   • Multiply both sides by 10: $\text{Opposite} = 0.5 \times 10$
   • $\text{Opposite} = 5$ units.

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

4. Key Takeaways

• Trigonometry studies relationships between angles and side lengths in triangles.
• Right-angled triangles are the starting point, having one 90-degree angle.
• The hypotenuse is always opposite the 90-degree angle and is the longest side.
• "Opposite" and "Adjacent" sides depend on which acute angle you're focusing on.
• SOH CAH TOA is your essential mnemonic for the sine, cosine, and tangent ratios.
• Use $\sin(\theta) = \text{Opposite} / \text{Hypotenuse}$ to relate opposite, hypotenuse, and angle.
• Use $\cos(\theta) = \text{Adjacent} / \text{Hypotenuse}$ to relate adjacent, hypotenuse, and angle.
• Use $\tan(\theta) = \text{Opposite} / \text{Adjacent}$ to relate opposite, adjacent, and angle.

Common mistakes to avoid:
- Mixing up which side is Opposite or Adjacent based on the chosen angle.
- Forgetting to use inverse functions ($\sin^{-1}$, etc.) when solving for an angle.
- Not checking that your calculator is in the correct mode (degrees vs. radians).
- Incorrectly identifying the hypotenuse (it's always opposite the right angle).

5. Now Try It

Draw a right-angled triangle. Label one of the acute angles as 45 degrees. If the side adjacent to the 45-degree angle is 7 cm long, use trigonometry to find the length of the hypotenuse.

What success looks like: You should be able to clearly identify the known angle, the known side, the unknown side, pick the correct trigonometric ratio, set up the equation, and solve for the hypotenuse's length. (Hint: $\cos(45^\circ) \approx 0.707$).