Know one side and one more part

The idea

If one side and any other part (an acute angle or another side) of a right triangle are known, the remaining sides and angles can be found using the trigonometric ratio that links the known and unknown parts.

A triangle with missing parts

A right triangle ABC has the right angle at B. You are told AB = 5 cm and angle C = 30°. That is all.

The sides BC and AC are missing. Trigonometry can find them from what you know, without measuring.

What is the least you must know to find all the missing sides and angles of a right triangle?

• One side and one more part: an acute angle or another side
• Only the angles
• Only one side

Choose the ratio

Each triangle is drawn to scale. Tap the ratio that contains the side you know and the part you want. A wrong ratio tells you what it is missing.

Link the known and the unknown

If one side and any other part (an acute angle or another side) of a right triangle are known, the remaining sides and angles can be found with the trigonometric ratio that links the known and the unknown parts.

Step 1: name the sides as opposite, adjacent and hypotenuse for the angle you use. Step 2: pick the ratio that has the side you know and the part you want. Step 3: solve. The Pythagoras theorem can also find a third side, as in Example 6 of your book.

Notes

Know one side and one more part of a right triangle? Pick the ratio that links the known part with the unknown part, and solve. The rest follows.

Check yourself

In triangle ABC, right-angled at B, AC = 8 cm and angle A = 30°. How long is BC (in cm)?

Answer: 4

sin 30° = BC / AC, so 1/2 = BC / 8 and BC = 4 cm.

Triangle PQR is right-angled at Q, with PQ = 4 cm and PR = 8 cm. Which ratio gives angle R?

In triangle ABC, right-angled at B, angle C = 60° and AC = 10 cm. How long is BC (in cm)?

Answer: 5

cos 60° = BC / AC, so 1/2 = BC / 10 and BC = 5 cm.

Which information is not enough to find the sides of a right triangle?

• cos R = 4/8. PQ is opposite to R, not adjacent. cos R uses the adjacent side QR.
• sin R = 4/8 — correct. Yes! PQ is opposite to R and PR is the hypotenuse, so sin R = PQ / PR = 4/8 = 1/2. That is 30°.
• tan R = 4/8. tan R divides by the adjacent side QR. PR is the hypotenuse.
• One side and one acute angle. This is enough. For example, the side and the angle link through sin, cos or tan.
• Two sides. This is enough. The Pythagoras theorem gives the third side, and a ratio gives the angles.
• Both acute angles, but no side — correct. Yes! Triangles of every size can have the angles 30° and 60°. Without one side, the size is not known.