Introduction to Trigonometry and Right-Angled Triangles

TL;DR

Trigonometry is all about finding unknown sides and angles in triangles, especially right-angled ones. The core concepts are the sine, cosine, and tangent ratios, which relate an angle to the lengths of two sides. You'll use these ratios to solve practical problems involving distances and angles.

1. The Mental Model

Imagine you're trying to figure out the height of a flagpole without climbing it. If you know how far you are from its base and the angle looking up to its top, trigonometry gives you the tools to calculate that height. It's like having a magic ruler for indirect measurements.

2. The Core Material

Trigonometry, at its heart, studies the relationships between the sides and angles of triangles. We'll start with right-angled triangles because they're the foundation. A right-angled triangle has one angle that measures exactly 90 degrees.

For any non-right angle in a right-angled triangle, we name the sides relative to that angle:
* Hypotenuse: This is always the longest side and is opposite the 90-degree angle.
* Opposite: This side is directly across from the angle you're interested in.
* Adjacent: This side is next to the angle you're interested in, but it's not the hypotenuse.

The Trigonometric Ratios (SOH CAH TOA)

Photo by Tomás Asurmendi on Pexels

There are three main ratios that link the angles to the side lengths:

• Sine (sin): The ratio of the length of the Opposite side to the length of the Hypotenuse.
  • sin(angle) = Opposite / Hypotenuse
• Cosine (cos): The ratio of the length of the Adjacent side to the length of the Hypotenuse.
  • cos(angle) = Adjacent / Hypotenuse
• Tangent (tan): The ratio of the length of the Opposite side to the length of the Adjacent side.
  • tan(angle) = Opposite / Adjacent

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

These ratios are constant for a given angle, no matter the size of the right-angled triangle. Your calculator knows these values.

Finding Missing Sides

Photo by Pixabay on Pexels

If you know one angle (other than the 90-degree one) and one side length, you can use these ratios to find a missing side.

1. Identify the known angle and the side you know.
2. Identify the side you want to find.
3. Choose the SOH, CAH, or TOA ratio that uses both the known side and the unknown side.
4. Set up the equation and solve for the unknown.

Finding Missing Angles

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

If you know two side lengths, you can use the inverse trigonometric functions to find a missing angle. These are often written as sin⁻¹, cos⁻¹, or tan⁻¹ (or arcsin, arccos, arctan) on your calculator.

1. Identify the two known sides relative to the angle you want to find.
2. Choose the SOH, CAH, or TOA ratio that uses those two sides.
3. Set up the equation with the angle as the unknown.
4. Use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) to find the angle.

graph TD
    A["Identify Right-Angled Triangle?"] --> B{{"Yes"}};
    A --> C{{"No"}};
    C --> D["Need other triangle rules (e.g., Sine/Cosine Rule)"];
    B --> E{"What do you know?"};
    E --> F["One Angle & One Side"];
    E --> G["Two Sides"];
    F --> H["Identify known/unknown sides relative to known angle"];
    H --> I{"Which ratio connects them?"};
    I --> J["Opposite & Hypotenuse"] --> K["Use Sine (SOH)"];
    I --> L["Adjacent & Hypotenuse"] --> M["Use Cosine (CAH)"];
    I --> N["Opposite & Adjacent"] --> O["Use Tangent (TOA)"];
    K --> P["Solve for missing side"];
    M --> P;
    O --> P;
    G --> Q["Identify known sides relative to unknown angle"];
    Q --> R{"Which ratio connects them?"};
    R --> S["Opposite & Hypotenuse"] --> T["Use arcsin (sin⁻¹)"];
    R --> U["Adjacent & Hypotenuse"] --> V["Use arccos (cos⁻¹)"];
    R --> W["Opposite & Adjacent"] --> X["Use arctan (tan⁻¹)"];
    T --> Y["Solve for missing angle"];
    V --> Y;
    X --> Y;

3. Worked Example

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

1. Known angle: 30 degrees.
2. Known side: Opposite = 5 units.
3. Unknown side: Hypotenuse (let's call it 'h').
4. Which ratio? We have Opposite and Hypotenuse, so we use Sine (SOH).
   • sin(angle) = Opposite / Hypotenuse
5. Set up the equation:
   • sin(30°) = 5 / h
6. Solve for h:
   • h * sin(30°) = 5
   • h = 5 / sin(30°)
   • From your calculator, sin(30°) = 0.5.
   • h = 5 / 0.5
   • h = 10

So, the hypotenuse is 10 units long.

4. Key Takeaways

• Trigonometry is used to find unknown sides or angles in triangles.
• Focus on right-angled triangles first, which have one 90-degree angle.
• The hypotenuse is always opposite the right angle and is the longest side.
• "SOH CAH TOA" helps you remember the sine, cosine, and tangent ratios.
• Use sin, cos, tan to find sides when an angle and a side are known.
• Use sin⁻¹, cos⁻¹, tan⁻¹ (inverse functions) to find angles when two sides are known.

Common Mistakes to Avoid

Photo by KATRIN BOLOVTSOVA on Pexels

• Mislabeling sides: Always correctly identify the Opposite, Adjacent, and Hypotenuse relative to the angle you're working with.
• Using the 90-degree angle: The trigonometric ratios (SOH CAH TOA) only apply to the acute angles (less than 90 degrees) in the right triangle, not the right angle itself.
• Calculator in wrong mode: Ensure your calculator is in "DEGREE" mode for most problems, not "RADIAN" mode, unless specified.
• Confusing ratios: Double-check that you're using the correct ratio (SOH, CAH, or TOA) for the sides and angle you have.

5. Now Try It

Draw a right-angled triangle. Label one of the non-90-degree angles as 40 degrees. Label the side adjacent to this 40-degree angle as 7 cm. Your task is to find the length of the hypotenuse and the side opposite the 40-degree angle.

Success looks like: You've correctly identified which trig ratios to use for each missing side, set up the equations correctly, and calculated both lengths to two decimal places.