Right Triangle Angles- Finding Angles with Two Sides

What You're Actually Working With

Every right triangle has three angles. One is always 90° — that's the right angle. The other two add up to 90° and are what you're trying to find when someone says "find the angles."

You don't need both of those angles. Find one, and you automatically know the other. That's the whole game here.

The Tool You Need: SOH CAH TOA

These three ratios tell you everything about right triangle angles:

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

Pick the right one based on which two sides you know. That's it. No memorization tricks, no fancy explanations.

Which Ratio to Use

This table makes it simple:

Quick Way to Remember

If you have the hypotenuse in your two sides, use sin or cos. If you don't have the hypotenuse, use tan.

How to Actually Calculate the Angle

Once you have the ratio, you need the inverse trig function. Your calculator has these: sin⁻¹, cos⁻¹, tan⁻¹ (also called arcsin, arccos, arctan).

The process:

1. Calculate the ratio (divide one side by the other)
2. Hit the inverse trig button for that ratio
3. Get your angle in degrees

Examples That Actually Make Sense

Example 1: You Know Opposite and Hypotenuse

Opposite side = 3, Hypotenuse = 5.

Step 1: Find the ratio → sin(θ) = 3/5 = 0.6

Step 2: Use inverse → θ = sin⁻¹(0.6)

Step 3: θ ≈ 36.87°

The other angle is 90° - 36.87° = 53.13°

Example 2: You Know Adjacent and Hypotenuse

Adjacent side = 8, Hypotenuse = 10.

Step 1: Find the ratio → cos(θ) = 8/10 = 0.8

Step 2: Use inverse → θ = cos⁻¹(0.8)

Step 3: θ ≈ 36.87°

The other angle is 53.13°

Example 3: You Know Opposite and Adjacent (No Hypotenuse)

Opposite = 4, Adjacent = 7.

Step 1: Find the ratio → tan(θ) = 4/7 ≈ 0.571

Step 2: Use inverse → θ = tan⁻¹(0.571)

Step 3: θ ≈ 29.74°

The other angle is 60.26°

Watch Out For These Mistakes

• Wrong ratio selection — always identify your hypotenuse first. It's always the longest side, across from the right angle.
• Forgetting to use inverse — sin(θ) gives you a ratio. sin⁻¹(ratio) gives you the angle. These are different buttons.
• Mixing up opposite and adjacent — opposite is across from the angle. Adjacent is next to the angle (not the hypotenuse).
• Calculator mode — make sure it's in degree mode, not radian mode, unless you're specifically solving in radians.

The 45-45-90 Shortcut

If the two legs (not the hypotenuse) are equal, you're looking at a 45-45-90 triangle. Both acute angles are 45°. No calculation needed.

When You Only Have One Ratio

Sometimes you only need one angle. You don't have to find both. The problem might only ask for one. Find the angle opposite your given side, calculate, and stop. You're done.

The second angle only matters if the problem asks for it or if you need it for the next step.