Welcome to the World of Right Triangles!

In this chapter, we are going to dive into Right Triangles and Trigonometry. While those words might sound a bit intimidating, think of right triangles as the "building blocks" of the geometric world. Whether you are measuring the height of a building or calculating the shortest path across a park, right triangles are everywhere!

For the SAT, mastering these concepts is like having a "cheat code." Many complex problems can be solved simply by spotting a hidden right triangle. Let’s break it down step-by-step.

1. The Foundation: The Pythagorean Theorem

Before we get into the fancy stuff, we need to know how the sides of a right triangle relate to each other. Every right triangle has two legs (the sides that make the \(90^\circ\) angle) and one hypotenuse (the long side across from the right angle).

The Formula: \(a^2 + b^2 = c^2\)

Where \(a\) and \(b\) are the legs, and \(c\) is always the hypotenuse.

Quick Tip: Pythagorean Triples

The SAT loves specific sets of whole numbers that always fit this formula. If you memorize these, you can save a lot of time calculating!

  • 3 - 4 - 5 (because \(3^2 + 4^2 = 5^2\))
  • 5 - 12 - 13
  • 8 - 15 - 17
  • 7 - 24 - 25

Note: Multiples of these also work! For example, a 6-8-10 triangle is just a 3-4-5 triangle that has been doubled.

Key Takeaway:

If you know two sides of a right triangle, you can always find the third using \(a^2 + b^2 = c^2\). Just make sure \(c\) is always the longest side!


2. Special Right Triangles

Don't worry if this seems tricky at first! There are two "famous" triangles that appear constantly on the SAT. These have specific ratios for their sides based on their angles.

The 45°-45°-90° Triangle (The Isosceles Right Triangle)

Think of this as a square cut in half diagonally. Because two angles are the same, two sides are the same.

  • Legs: \(x\)
  • Hypotenuse: \(x\sqrt{2}\)

Analogy: If the leg is a "step," the hypotenuse is just that "step" multiplied by \(\sqrt{2}\).

The 30°-60°-90° Triangle

Think of this as an equilateral triangle cut in half. The sides follow a very specific pattern:

  • Short Leg (across from 30°): \(x\)
  • Long Leg (across from 60°): \(x\sqrt{3}\)
  • Hypotenuse (across from 90°): \(2x\)
Memory Aid:

In a 30-60-90 triangle, the Hypotenuse is always double the Short Leg. It’s as easy as \(1, 2, 3\) (specifically \(1, \sqrt{3}, 2\)).


3. Right Triangle Trigonometry (SOH CAH TOA)

Trigonometry is just a fancy way of talking about the ratios between sides. We use three main functions: Sine, Cosine, and Tangent.

The Mnemonic: SOH CAH TOA

This is the most famous memory aid in math! Use it to remember which sides to divide:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Step-by-Step: How to find a Ratio

  1. Pick the angle you are looking from (let's call it angle \(\theta\)).
  2. Identify the Opposite side (the one furthest away).
  3. Identify the Hypotenuse (the long diagonal side).
  4. Identify the Adjacent side (the side right next to your angle that isn't the hypotenuse).
  5. Plug them into your SOH CAH TOA fraction!
Common Mistake to Avoid:

Never use the \(90^\circ\) angle as your starting point for SOH CAH TOA. Always use one of the two smaller angles!


4. The Sine and Cosine Relationship

Did you know? The Sine of one angle in a right triangle is always equal to the Cosine of the other acute angle. This is because those two angles always add up to \(90^\circ\) (they are complementary).

The Formula: \(\sin(x^\circ) = \cos(90^\circ - x^\circ)\)

Example: \(\sin(20^\circ)\) is the exact same value as \(\cos(70^\circ)\).

Key Takeaway:

If an SAT question says \(\sin(A) = \cos(B)\), you immediately know that \(A + B = 90\)!


5. The Unit Circle and Radians

Sometimes the SAT will move from triangles into a circle. The Unit Circle is just a circle with a radius of \(1\).

What are Radians?

Radians are just another way to measure angles, like Celsius vs. Fahrenheit. Instead of degrees, we use \(\pi\).

  • The Conversion: \(180^\circ = \pi\) radians
  • To turn degrees into radians: Multiply by \(\frac{\pi}{180}\)
  • To turn radians into degrees: Multiply by \(\frac{180}{\pi}\)

Coordinates on the Circle

If you have a point on the unit circle at a certain angle \(\theta\), the coordinates \((x, y)\) are actually:
\(x = \cos(\theta)\)
\(y = \sin(\theta)\)

Quick Review:

On the unit circle, Cosine is the x-value and Sine is the y-value. Think alphabetical order: \(C\) (Cosine) comes before \(S\) (Sine), just like \(x\) comes before \(y\)!


Summary Checklist for Success

Before you tackle the practice problems, ask yourself:

  • Can I identify the hypotenuse? (It’s always across from the box/right angle).
  • Do I remember SOH CAH TOA?
  • Do I know my 30-60-90 and 45-45-90 patterns?
  • Do I remember that \(\sin(x) = \cos(90-x)\)?

You've got this! Trigonometry on the SAT isn't about being a math genius; it's about recognizing these patterns and applying the right tool for the job.