Solving Right Triangles

Find missing sides and angles in right triangles using trigonometry. Math 10-C Alberta curriculum.

What you'll learn in this lesson

The Strategy for Solving Triangles. Learn what it means to fully solve a right triangle and master the three-step strategy: label the sides, identify knowns and unknowns, and choose the right SOH CAH TOA ratio for every problem.
Solving a Triangle from Three Sides. Work through a complete example where all three sides are given and you need to find the missing angles using inverse tangent. Learn the angle-sum verification shortcut and the Pythagorean check.
Solving a Triangle from One Side and One Angle. Given only the hypotenuse and one acute angle, use sine and cosine to find both missing sides, then apply the verification check to confirm your answer.
Finding Specific Variables. Practice finding a single unknown side or angle without solving the entire triangle. Work through two focused examples involving the hypotenuse, a horizontal side, and an unknown angle.

Definition — Solving a Right Triangle. To solve a right triangle means to find the measures of all three angles and all three sides.

You start with some known information and use trig ratios and the Pythagorean theorem to fill in everything else.

Every trig problem follows the same three-step approach.

Step 1 — Label the Sides. Relative to the acute angle you are working with, identify which side is:

Opposite — the side across from the angle.

Adjacent — the side next to the angle (not the hypotenuse).

Hypotenuse — the longest side, always opposite the right angle.

Step 2 — Identify What You Know and What You Need. You will always have at least one side and one angle, or two sides.

Decide which unknown you want to find before choosing a ratio.

Step 3 — Choose the Right Ratio Using SOH CAH TOA. Look at the two sides that are involved — the one you know and the one you want.

The ratio that connects those two sides is the one you need.

The third side is irrelevant for that step.

Pro Tip. Whenever you are unsure which ratio to use, ask: "Which two sides am I working with?"

Opposite and Hypotenuse? Use sin (SOH).

Adjacent and Hypotenuse? Use cos (CAH).

Opposite and Adjacent? Use tan (TOA).

When all three sides are already known, you only need to find the two missing angles.

Given Information. Right angle at B.

AB = 12 cm (horizontal, bottom side).

BC = 5 cm (vertical, right side).

AC = 13 cm (diagonal hypotenuse).

Find C. Step 1 — Label from C:

Opposite = AB = 12 cm.

Adjacent = BC = 5 cm.

Step 2 — Choose the ratio:

Opposite and Adjacent are known, so use .

Step 3 — Set up and solve:

C = (12)/(5)

C = ^(-1)(2.4)

Answer: C ≈ 67.4°

Find A. Step 1 — Use the angle sum:

The three angles of a triangle always add to 180°.

Step 2 — Subtract the known angles:

A = 180° - 90° - 67.4°

Answer: A ≈ 22.6°

Verification Shortcut. Angle check: 22.6° + 90° + 67.4° = 180° ✓

Side check (Pythagorean theorem): 5^2 + 12^2 = 25 + 144 = 169 = 13^2 ✓

Both checks pass — the triangle is fully solved.

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