Trigonometry

TL;DR

Trigonometry helps you find missing sides or angles in right-angled triangles using special ratios involving sine, cosine, and tangent. You'll learn to identify the sides of a right-angled triangle relative to a given angle and then pick the correct ratio to solve for what's unknown. This is super useful in real-world problems involving heights, distances, and angles.

1. The Mental Model

Imagine you have a right-angled triangle, and you know some parts but not others. Trigonometry is like having a toolkit that lets you figure out those missing parts (sides or angles) just by knowing how the sides relate to the angles inside that specific type of triangle.

2. The Core Material

Trigonometry focuses on the relationships between the angles and sides of right-angled triangles. A right-angled triangle has one angle that is exactly 90 degrees.

First, let's name the sides of a right-angled triangle relative to a reference angle (the angle you're working with, which isn't the 90-degree one).

• Hypotenuse: This is always the longest side and is directly opposite the right angle.
• Opposite: This side is directly across from your reference angle.
• Adjacent: This side is next to your reference angle and isn't the hypotenuse.

The Trigonometric Ratios (SOH CAH TOA)

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There are three main trigonometric ratios: Sine (sin), Cosine (cos), and Tangent (tan). You can remember them with the mnemonic SOH CAH TOA:

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

Here, $\theta$ (theta) is the symbol we often use for the reference angle.

How to use the ratios to find a missing side

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1. Identify: Label the hypotenuse, opposite, and adjacent sides relative to your known angle.
2. Choose: Look at what you know (an angle and one side) and what you want to find (another side). Pick the SOH, CAH, or TOA ratio that uses these three parts.
3. Substitute: Plug your known values into the chosen ratio formula.
4. Solve: Rearrange the formula to find the unknown side.

How to use the ratios to find a missing angle

Photo by RDNE Stock project on Pexels

1. Identify: Label the hypotenuse, opposite, and adjacent sides relative to the unknown angle.
2. Choose: Look at the two sides you know. Pick the SOH, CAH, or TOA ratio that uses those two sides.
3. Substitute: Plug your known side values into the chosen ratio formula.
4. Solve: To find the angle, you'll need to use the inverse trigonometric functions:
   • If sin($\theta$) = ratio, then $\theta$ = sin⁻¹(ratio)
   • If cos($\theta$) = ratio, then $\theta$ = cos⁻¹(ratio)
   • If tan($\theta$) = ratio, then $\theta$ = tan⁻¹(ratio)
     You'll find these (sin⁻¹, cos⁻¹, tan⁻¹) buttons on your calculator, usually by pressing "shift" or "2nd F" before the sin, cos, or tan button.

Here's a diagram showing the decision process:

graph TD
    A["Start: You have a Right-Angled Triangle"] --> B{"What do you want to find?"}
    B --> C["Missing Side"]
    B --> D["Missing Angle"]

    C --> E{"What do you know?"}
    E --> F["Angle and Adjacent Side"]
    E --> G["Angle and Opposite Side"]
    E --> H["Angle and Hypotenuse"]

    F --> I{"Want Opposite? Use Tan"}
    F --> J{"Want Hypotenuse? Use Cos"}

    G --> K{"Want Adjacent? Use Tan"}
    G --> L{"Want Hypotenuse? Use Sin"}

    H --> M{"Want Opposite? Use Sin"}
    H --> N{"Want Adjacent? Use Cos"}

    D --> O{"What two sides do you know?"}
    O --> P["Opposite & Hypotenuse"]
    O --> Q["Adjacent & Hypotenuse"]
    O --> R["Opposite & Adjacent"]

    P --> S["Use sin⁻¹(Opposite / Hypotenuse)"]
    Q --> T["Use cos⁻¹(Adjacent / Hypotenuse)"]
    R --> U["Use tan⁻¹(Opposite / Adjacent)"]

3. Worked Example

Let's find the missing side, x, in a right-angled triangle.

Problem: You have a right-angled triangle. One angle is 35 degrees. The side adjacent to this angle is 10 cm. You want to find the side opposite the 35-degree angle.

1. Identify:

   • The 90-degree angle is known.
   • The reference angle ($\theta$) is 35 degrees.
   • The side we know is 10 cm, which is adjacent to the 35-degree angle.
   • The side we want to find is x, which is opposite the 35-degree angle.
   • We don't know or need the hypotenuse for this problem.

2. Choose: We have the Opposite side (x) and the Adjacent side (10 cm). The ratio that uses Opposite and Adjacent is TOA (Tan = Opposite / Adjacent).

3. Substitute:
   tan(35°) = x / 10

4. Solve:
   To get x by itself, multiply both sides by 10:
   x = 10 * tan(35°)

   Using a calculator (make sure it's in "degree" mode!):
   tan(35°) ≈ 0.7002
   x = 10 * 0.7002
   x ≈ 7.002 cm

   So, the missing side x is approximately 7.00 cm (to 3 significant figures).

4. Key Takeaways

• Trigonometry is used for finding unknown sides or angles in right-angled triangles.
• Always correctly label the Hypotenuse, Opposite, and Adjacent sides relative to your reference angle.
• Remember SOH CAH TOA to pick the right trigonometric ratio (Sine, Cosine, or Tangent).
• To find a missing angle, you'll need to use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) on your calculator.
• Your calculator must be in degree mode for these calculations to be correct.
• Trigonometry has many real-world applications, from construction to navigation.

Common Mistakes to Avoid:
- Mixing up sides: Accidentally swapping the opposite and adjacent sides is a very common error. Always draw and label!
- Wrong ratio: Picking the wrong SOH CAH TOA ratio for the given information. Double-check your choice.
- Calculator mode: Forgetting to switch your calculator to "degree" mode instead of "radian" or "gradian" mode, leading to completely wrong answers.
- Inverse for sides: Trying to use sin⁻¹ to find a side length instead of just sin, cos, or tan. Inverse functions are only for finding angles.

5. Now Try It

Grab a piece of paper and your calculator. Draw a right-angled triangle. Label the right angle. Pick one of the other angles and label it 40 degrees. Label the side adjacent to the 40-degree angle as 8 metres. Your task is to find the length of the hypotenuse.

What to do:
1. Draw the triangle and clearly label the 40-degree angle, the 8m adjacent side, and the hypotenuse (which you're trying to find).
2. Decide which trigonometric ratio (SOH, CAH, or TOA) connects the 40-degree angle, the adjacent side, and the hypotenuse.
3. Write down the equation using the chosen ratio.
4. Solve the equation for the hypotenuse length.
5. Make sure your calculator is in degree mode!

What success looks like:
You should get a hypotenuse length of approximately 10.4 metres (to 3 significant figures). If you get this, you've successfully applied the CAH ratio.