Solving Right Triangles

Part 1 of 2: Foundation

In this lesson:

• Define what it means to "solve" a right triangle
• Use trig ratios and inverse trig to find unknowns
• Combine tools strategically

Learning Objectives

By the end of this lesson, you should be able to:

1. Define what it means to "solve a right triangle"
2. Use trig ratios to find unknown sides when you know an angle
3. Use inverse trig to find unknown angles when you know two sides
4. Apply the Pythagorean Theorem strategically alongside trig
5. Choose the right tool for each situation

Quick Review

You already know these tools:

• SOH-CAH-TOA - connects angles to side ratios
• Pythagorean Theorem - connects the three sides

Today we'll learn when to use each one - and how to use them together.

What Does "Solve a Right Triangle" Mean?

"Solve" means find all unknown sides and angles

Minimum Information Needed

To solve a right triangle, you need:

• One side + one acute angle, OR
• Two sides

You cannot solve with only angles.

The 5-Step Process

Every time you solve a right triangle, follow these five steps

The Five Steps in Detail

1. Identify what you know
2. List what you need to find
3. Choose the right tool for each unknown
4. Calculate carefully
5. Check your work

Check-In

What minimum information do you need to solve a right triangle?

Think about why angles alone aren't enough...

Example 1: One Side and One Angle

Given: leg = 7, angle = 40° (opposite the known leg)

Find: the other two sides and the other acute angle

Example 1: Label O, A, H

Label opposite, adjacent, and hypotenuse relative to the angle

Example 1: Choose Tools

Step 3: Choose the right tool for each unknown

• For hypotenuse: - connecting O to H
• For adjacent leg: - connecting O to A
• For the other angle:

Example 1: Calculate

Step 4: Calculate each unknown

Hypotenuse:

Adjacent leg:

Other angle:

Example 1: Check Your Work

Step 5: Check

Do the angles add to 180°?

Does Pythagorean hold?

Example 2: Another Setup

Given: hypotenuse = 10, angle = 35°

Solution:

• Other angle:
• Opposite leg: →
• Adjacent leg: →

Check-In

Did we find ALL unknowns?

What if we stopped after finding the two legs?

What would still be missing?

Transition: Combining Tools

Now let's combine trig and Pythagorean strategically

Pattern 1: Angle + One Side

When you know an angle and one side:

• Use trig to find the other two sides directly
• Pythagorean is a check, not the main tool

Pattern 1 Example: Hypotenuse and Angle

Given: hypotenuse = 15, angle = 60°

Find both legs:

• Opposite: →
• Adjacent: →

Pattern 1: Alternative Check

Check with Pythagorean:

Small rounding difference - both methods work!

Check-In

Which trig ratio connects opposite and hypotenuse?

Think about SOH-CAH-TOA...

Pattern 2: Two Sides Only

When you know two sides but no angles:

• Use Pythagorean to find the third side
• Use inverse trig to find the angles

Pattern 2 Example: Two Legs

Given: legs 6 and 8

Step 1: Find hypotenuse with Pythagorean

Pattern 2 Example: Find Angles

Step 2: Find angles with inverse trig

Now we have all five unknowns!

Check-In

Can Pythagorean Theorem find an angle?

Why not?

Tool Selection Table

Quick reference for choosing tools

Your Turn: Practice

Given: leg = 5, hypotenuse = 12

Your task:

1. Identify which pattern this fits
2. Find the other leg
3. Find both acute angles

Try each step before checking your answer

Summary: Foundation Complete

✓ Solve = find all sides and angles
✓ 5-step process: identify, list, choose, calculate, check
✓ Pattern 1: angle + side → trig first
✓ Pattern 2: two sides → Pythagorean, then inverse trig

Next: Part 2 applies these tools to real-world problems