Right-triangle trigonometry and the Pythagorean theorem (Geometry) · lesson

The Pythagorean theorem

💡 In a right triangle, the two legs squared add up to the hypotenuse squared: a² + b² = c².

The hypotenuse is the longest side. It sits across from the right angle. The other two sides are the legs.

To find the hypotenuse, square both legs, add, and take the square root. To find a leg, square the hypotenuse, subtract the other leg squared, and take the square root.

🧠 Let's work it out

A right triangle has a hypotenuse of 25 and a leg of 7. Find the other leg.

Use a² + b² = c²: b² = 25² − 7² = 625 − 49 = 576.
Take the root: b = √576 = 24.

What do you think comes next?

hypotenuse: the side across from the right angle, the longest side
leg: one of the two sides that form the right angle

🔗 Khan Academy: Right triangles & trigonometry · Khan Academy (free) ↗

Special right triangles

💡 In a 30-60-90 triangle the hypotenuse is twice the shortest side; in a 45-45-90 triangle it is a leg times √2.

A 30-60-90 triangle is half of an equilateral triangle. Its sides are in the ratio 1 : √3 : 2. The shortest side is across from the 30° angle.

A 45-45-90 triangle has two equal legs. Its sides are in the ratio 1 : 1 : √2.

🧠 Let's work it out

In a 30-60-90 triangle, the side across from 30° is 5. Find the other two sides.

That is the shortest side.
The hypotenuse is 2 × 5 = 10.
The long leg is 5√3, about 8.66.

What do you think comes next?

equilateral: having all sides equal

Sine, cosine, and tangent find sides

💡 Pick the ratio that uses the side you know and the side you want: SOH CAH TOA.

Stand at an angle that is not the right angle. The opposite side is across from it. The adjacent side touches it and is not the hypotenuse.

Sine = opposite ÷ hypotenuse. Cosine = adjacent ÷ hypotenuse. Tangent = opposite ÷ adjacent. The memory trick is SOH CAH TOA.

Set up the ratio, then multiply or divide to get the side alone. Keep your calculator in degree mode.

🧠 Let's work it out

One angle is 25° and the hypotenuse is 30. Find the side across from 25°.

Opposite and hypotenuse go with sine.
Set it up: sin 25° = opposite ÷ 30.
So opposite = 30 × sin 25°.
30 × 0.4226 ≈ 12.68, which rounds to 12.7.

What do you think comes next?

sine: opposite side ÷ hypotenuse
cosine: adjacent side ÷ hypotenuse
tangent: opposite side ÷ adjacent side
adjacent: next to; the leg that touches the angle

Inverse trig finds angles

💡 When you know two sides and want the angle, use sin⁻¹, cos⁻¹, or tan⁻¹.

The trig keys turn an angle into a ratio. The inverse keys go the other way: they turn a ratio into an angle.

Pick the ratio from the two sides you know. Divide them, then use the matching inverse key.

🧠 Let's work it out

A right triangle has legs 9 and 14. Find the angle across from the side of 9.

Opposite and adjacent go with tangent: tan θ = 9/14.
Undo the tangent: θ = tan⁻¹(9 ÷ 14) ≈ 32.74°.

What do you think comes next?

inverse tangent: tan⁻¹, the key that finds an angle from opposite ÷ adjacent

Angles of elevation and depression

💡 Draw the right triangle first: the height is opposite the angle and the ground distance is adjacent.

The angle of elevation is how far you look up from flat. The angle of depression is how far you look down from flat. Looking down from a tower to a boat makes the same angle as looking up from the boat.

If your eyes are above the ground, the triangle starts at eye level. Add your eye height at the end.

With two angles from two spots, call the height h. Write each ground distance as h ÷ tan(angle). The two distances differ by how far you walked.

🧠 Let's work it out

From the top of an 80-foot cliff, the angle of depression to a boat is 20°. How far is the boat from the cliff?

The height 80 is opposite the angle. The distance d is adjacent.
So tan 20° = 80 ÷ d.
Solve for d: d = 80 ÷ tan 20° ≈ 80 ÷ 0.364 ≈ 219.8.

What do you think comes next?

angle of elevation: the angle you look up from a flat line
angle of depression: the angle you look down from a flat line