Introduction to Trigonometry: Ratios and Applications

TL;DR

Trigonometry is about the relationships between the sides and angles of right-angled triangles. You'll learn three main ratios – sine, cosine, and tangent – to find unknown sides or angles. These ratios are super useful in fields like engineering, physics, and even video game development.

1. The Mental Model

Imagine you have a magic toolkit that lets you measure distances and heights without a ruler, just by knowing some angles and one side. That's essentially what trigonometry does for right-angled triangles; it's a way to unlock information about their hidden parts.

2. The Core Material

Trigonometry focuses on right-angled triangles (triangles with one 90-degree angle). The key idea is that for any given angle (other than the 90-degree one) in a right triangle, the ratio of its sides will always be the same, no matter how big or small the triangle is.

2.1 Identifying Sides

Before you can use the ratios, you need to correctly label the sides of your right-angled triangle relative to a specific angle you're interested in (we'll call this angle $\theta$ - "theta"):

• Hypotenuse: Always the longest side, and it's always opposite the 90-degree angle.
• Opposite: The side directly across from the angle $\theta$.
• Adjacent: The side next to the angle $\theta$ that is not the hypotenuse.

2.2 The Three Core Ratios (SOH CAH TOA)

There are three primary trigonometric ratios: Sine, Cosine, and Tangent. A common mnemonic to remember them is SOH CAH TOA:

• SOH: Sine ($\sin$) = Opposite / Hypotenuse
• CAH: Cosine ($\cos$) = Adjacent / Hypotenuse
• TOA: Tangent ($\tan$) = Opposite / Adjacent

These ratios are functions that take an angle as input and give you a ratio as output. Your calculator has buttons for these!

2.3 Finding Unknown Sides

If you know one angle and one side of a right-angled triangle, you can use these ratios to find an unknown side.

Example: You have a 30-degree angle, and the hypotenuse is 10 units long. You want to find the length of the opposite side.

1. Identify: You know the Hypotenuse and the angle. You want the Opposite side.
2. Choose Ratio: SOH (Sine = Opposite/Hypotenuse) involves Opposite and Hypotenuse.
3. Set up: $\sin(30^\circ) = \text{Opposite} / 10$
4. Solve: $\text{Opposite} = 10 \times \sin(30^\circ) = 10 \times 0.5 = 5$ units.

2.4 Finding Unknown Angles

If you know two sides of a right-angled triangle, you can use the inverse trigonometric functions (often denoted as $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ on your calculator) to find an unknown angle.

Example: The opposite side is 5 units, and the adjacent side is 5 units. You want to find the angle.

1. Identify: You know Opposite and Adjacent. You want the angle.
2. Choose Ratio: TOA (Tangent = Opposite/Adjacent) involves Opposite and Adjacent.
3. Set up: $\tan(\theta) = 5 / 5 = 1$
4. Solve: $\theta = \tan^{-1}(1) = 45^\circ$.

graph TD
    A["Identify Right-Angled Triangle"] --> B["Identify REFERENCE Angle (θ)"];
    B --> C1{"Know θ & One Side?"};
    B --> C2{"Know Two Sides?"};

    C1 -- Yes --> D1["Label Sides relative to θ:
    - Hypotenuse
    - Opposite
    - Adjacent"];
    C2 -- Yes --> D1;

    D1 --> E1{"Want Unknown Side?"};
    D1 --> E2{"Want Unknown Angle?"};

    E1 -- Yes --> F1["Choose SOH CAH TOA Ratio that
    includes Known Side(s) & Unknown Side"];
    E2 -- Yes --> F2["Choose SOH CAH TOA Ratio that
    includes Known Two Sides"];

    F1 --> G1["Set up Equation (e.g., sin(θ) = Opp/Hyp)"];
    F2 --> G2["Set up Equation (e.g., tan(θ) = Opp/Adj)"];

    G1 --> H1["Use Calculator (sin, cos, tan)
    to Solve for Unknown Side"];
    G2 --> H2["Use Inverse Calculator (sin⁻¹, cos⁻¹, tan⁻¹)
    to Solve for Unknown Angle"];

3. Worked Example

You're standing 20 meters away from the base of a tree. You look up at the top of the tree, and the angle of elevation (the angle from your horizontal line of sight upwards) is 35 degrees. How tall is the tree?

1. Draw it: Sketch a right-angled triangle. Your position is one vertex, the base of the tree is the 90-degree corner, and the top of the tree is the third vertex.
2. Label:
   • The angle of elevation (from you to the top of the tree) is $35^\circ$. This is your $\theta$.
   • The distance from you to the tree is 20 meters. Relative to your $35^\circ$ angle, this is the Adjacent side.
   • The height of the tree is what you want to find. Relative to your $35^\circ$ angle, this is the Opposite side.
   • The hypotenuse isn't needed right now.
3. Choose Ratio: You know the Adjacent side and want the Opposite side. The ratio that connects Opposite and Adjacent is TOA (Tangent = Opposite/Adjacent).
4. Set up the equation:
   $\tan(35^\circ) = \text{Opposite} / \text{Adjacent}$
   $\tan(35^\circ) = \text{Height} / 20$
5. Solve for the unknown:
   $\text{Height} = 20 \times \tan(35^\circ)$
   Using a calculator, $\tan(35^\circ) \approx 0.7002$.
   $\text{Height} \approx 20 \times 0.7002$
   $\text{Height} \approx 14.004$ meters.

So, the tree is approximately 14 meters tall.

4. Key Takeaways

• Trigonometry only works directly with right-angled triangles.
• The three main ratios (sine, cosine, tangent) relate angles to side ratios.
• SOH CAH TOA is a crucial memory aid for the ratios.
• Always correctly identify the hypotenuse, opposite, and adjacent sides relative to the angle you're using.
• Use sine, cosine, or tangent to find a side when you know an angle and another side.
• Use inverse sine ($\sin^{-1}$), inverse cosine ($\cos^{-1}$), or inverse tangent ($\tan^{-1}$) to find an angle when you know two sides.
• Ensure your calculator is in "degree" mode for these problems.

Common Mistakes to Avoid

• Confusing Opposite and Adjacent: This is the most common error; always label your sides from the perspective of your chosen angle.
• Using the 90-degree angle for ratios: The SOH CAH TOA ratios only apply to the other two acute (less than 90-degree) angles.
• Calculator Mode: Forgetting to switch your calculator between "degrees" and "radians" mode if you're not careful (for introductory trig, you'll almost always use degrees).
• Mixing up regular and inverse functions: Use sin/cos/tan to find a ratio from an angle, and $\sin^{-1}/\cos^{-1}/\tan^{-1}$ to find an angle from a ratio.

5. Now Try It

You have a ladder that is 8 meters long. You lean it against a wall, and the base of the ladder is 2 meters away from the wall. What angle does the ladder make with the ground? (Assume the wall is perfectly vertical, forming a right-angled triangle).

What to do:
1. Draw the scenario and label the knowns (ladder length, distance from wall).
2. Identify which sides correspond to the opposite, adjacent, and hypotenuse for the angle you want to find (the angle with the ground).
3. Choose the correct trigonometric ratio (SOH, CAH, or TOA).
4. Set up the equation.
5. Use your calculator's inverse trigonometric function to find the angle.

What success looks like: You should get an angle around 75.5 degrees.