Right Triangle Trigonometry

This explains why $\sin 30^\circ = \cos 60^\circ$. They are two sides of the exact same geometric coin.

🗺️ What You'll Master

📐 Part 1: Mastering the Right Triangle & Pythagorean Theorem

1.1 Anatomy of a Right Triangle

Every right triangle has three sides with special names, and which side is "opposite" or "adjacent" depends on the angle you're looking at (we call it $\theta$):

Let's compare these three crucial sides to see how they behave depending on your perspective:

1.2 The Pythagorean Theorem

1.3 The Euclidean Theorem (Geometric Mean Theorems)

When you drop an altitude from the right angle of a right triangle to its hypotenuse, you create three similar triangles. This unlocks powerful proportional relationships known as the Euclidean Theorem.

Let $\triangle ABC$ be a right triangle at $A$, and let $AD$ be the altitude drawn to the hypotenuse $BC$.

These formulas provide a brilliant shortcut for finding missing lengths without calculating any angles.

🔑 Part 2: Unlocking the Three Basic Ratios (SOH-CAH-TOA)

2.1 Sine, Cosine & Tangent

The three fundamental trigonometric ratios connect the sides of a right triangle to its angles:

To memorize these formulas easily, mathematicians created a famous mnemonic that you can rely on during exams:

2.2 Finding Missing Sides Using Trig Ratios

2.3 Finding Missing Angles: Inverse Trig Functions

🧩 Part 3: Discovering the Six Trigonometric Functions

3.1 Meet the Full Family

Beyond the Big Three (sin, cos, tan), there are three more functions, each is simply the reciprocal (flip) of one of the originals:

Learning these new functions can be confusing at first, especially because of the tricky names. Here is a helpful tip to keep them straight:

⭕ Part 4: Mastering the Unit Circle

4A: What Is the Unit Circle?

Take a look at the basic structure of this circle on the coordinate plane:

Why radius = 1? Because it makes everything beautifully simple. Watch:

The Golden Connection

Why does this work? Draw a right triangle inside the unit circle:

Since the hypotenuse (radius) = 1: $\sin \theta = \frac{\text{opp}}{\text{hyp}} = \frac{\text{opp}}{1} = \text{opp} = y$ $\cos \theta = \frac{\text{adj}}{\text{hyp}} = \frac{\text{adj}}{1} = \text{adj} = x$

And from these, all six functions on the unit circle:

4B: Famous Angles & Patterns

These five angles appear constantly. You must know them by heart:

Also know the axis angles:

4C: The Four Quadrants (Where Signs Live)

Here is how those four quadrants are laid out visually:

Depending on which quadrant your angle lands in, the trigonometric functions will be either positive or negative. Here is the complete breakdown:

🕵️‍♂️ Part 5: Decoding Reference Angles

5.1 What Is a Reference Angle?

5.2 How to Calculate the Reference Angle

5.3 The Master Method: Three Steps to Any Trig Value

🧭 Part 6: Navigating Directed & Co-terminal Angles

6.1 Directed Angles

Let's visualize how these positive and negative rotations look on the coordinate plane:

6.2 Co-terminal Angles

To find these mathematically equivalent angles, you can use a simple addition or subtraction process:

6.3 Key Skills with Co-terminal Angles

⚔️ Part 7: Conquering the Sine & Cosine Laws

7.1 Beyond Right Triangles

Everything so far required a right triangle (one 90° angle). But real life is messy, most triangles in architecture, surveying, and navigation don't have a right angle. For these oblique triangles, we need two powerful laws.

7.2 The Law of Sines (Sine Law)

When to use Sine Law:

To see exactly how this ambiguous case plays out in a real problem, follow this step-by-step example:

7.3 Deriving the Sine Law

7.4 The Law of Cosines (Cosine Law)

This law might look complex, but it contains a brilliant hidden secret that connects it directly back to the very first concept we learned:

When to use Cosine Law:

7.5 Deriving the Cosine Law

7.6 The Decision Flowchart: Which Law Do I Use?

🏗️ Part 8: Applying Trigonometry to the Real World (Elevation & Depression)

8.1 Trigonometry Meets the Real World

This is where trigonometry stops being abstract and starts solving problems you can see with your own eyes, from measuring building heights to calculating flight paths.

🏡 Capstone Connection: Sustainable Houses

🌍 The Bigger Picture

You have just mastered the exact mathematical tools that ancient civilizations used to navigate oceans and build architectural wonders. You didn't just memorize formulas; you learned a mathematical framework to decode the physical world. Whether you're aiming for engineering, physics, or computer science, you now have the tools to model and solve complex real-world problems. Keep building on this foundation!

📖 Vocabulary

⏳ 30-Second Exam Memory Triggers

🧾 Formula Summary

Right Triangles

• Pythagorean Theorem: $a^2 + b^2 = c^2$
• Primary Ratios: $\sin \theta = \frac{\text{Opp}}{\text{Hyp}}$, $\cos \theta = \frac{\text{Adj}}{\text{Hyp}}$, $\tan \theta = \frac{\text{Opp}}{\text{Adj}}$
• Reciprocal Ratios: $\csc \theta = \frac{\text{Hyp}}{\text{Opp}}$, $\sec \theta = \frac{\text{Hyp}}{\text{Adj}}$,
• Euclidean Altitude Rule: $AD^2 = BD \times CD$
• Euclidean Leg Rule: $AB^2 = BD \times BC$, $AC^2 = CD \times BC$
• Complementary Angles: $\sin \theta = \cos(90^\circ - \theta)$, $\cos \theta = \sin(90^\circ - \theta)$,

The Unit Circle

• Unit Circle Coordinates: $(x, y) = (\cos \theta, \sin \theta)$

Oblique Triangles

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines: $a^2 = b^2 + c^2 - 2bc\cos A$
• Co-terminal Angles: $\theta_{\text{new}} = \theta \pm 360^\circ n$

🎯 Key Takeaways

📝 Practice Questions

Watch out! The names are deceptive: Cosecant is the reciprocal of Sine (not cosine!).

Notice: each pair has the same two numbers, just flipped!

$\boxed{\cos 240^\circ = -\frac{1}{2}}$

$\boxed{\tan 315^\circ = -1}$

$\boxed{\sin 330^\circ = -\frac{1}{2}}$