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Amy Ferguson Moncure

IB AA · Unit 2 · 5B Rules of Trigonometry

Right Triangles

Use sine, cosine, and tangent in a right triangle to find an angle or a side.

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Key Ideas Easy Medium Hard

sin θ = opphyp

cos θ = adjhyp

tan θ = oppadj

What to Remember

Name the sides from the angle

The opposite side is across from the angle you are using.

The adjacent side is the other leg.

The hypotenuse is across from the right angle.

The three ratios

sin θ = oppositehypotenuse

cos θ = adjacenthypotenuse

tan θ = oppositeadjacent

Find an angle

Write the ratio, substitute, then use sin−1, cos−1, or tan−1.

Give the angle to 1 decimal place unless it is exact.

Find a side

Write the ratio that uses the known angle and the known side.

Give a decimal side to 3 significant figures.

Easy

Find the angle

Right-angled at C. BC is opposite angle A, and AC is adjacent to angle A.

Example
Triangle ABC is right-angled at C. BC = 12 cm and AC = 9 cm. Find angle A to 1 decimal place.

tan A = oppositeadjacent

tan A = 129

A = 53.1°

Answer: 53.1°

1
Triangle ABC is right-angled at C. BC = 8 cm and AC = 15 cm. Find angle A.
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tan A = oppositeadjacent

tan A = 815

A = 28.1°

Answer: 28.1°

2
Triangle ABC is right-angled at C. BC = 7 cm and AC = 4 cm. Find angle A.
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tan A = oppositeadjacent

tan A = 74

A = 60.3°

Answer: 60.3°

3
Triangle ABC is right-angled at C. BC = 20 cm and AC = 21 cm. Find angle A.
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tan A = oppositeadjacent

tan A = 2021

A = 43.6°

Answer: 43.6°

4
Triangle ABC is right-angled at C. BC = 5 cm and AC = 12 cm. Find angle A.
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tan A = oppositeadjacent

tan A = 512

A = 22.6°

Answer: 22.6°

5
Triangle ABC is right-angled at C. BC = 9 cm and AC = 40 cm. Find angle A.
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tan A = oppositeadjacent

tan A = 940

A = 12.7°

Answer: 12.7°

Medium

Find a leg

Match the ratio to the side you know and the side you need.

Example
Triangle ABC is right-angled at C. Angle A = 38° and hypotenuse AB = 20 cm. Find BC.

sin 38° = oppositehyp

opposite = 20 × sin 38°

opposite = 12.3 cm

Answer: 12.3 cm

1
Triangle ABC is right-angled at C. Angle A = 42° and AB = 16 cm. Find BC.
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sin 42° = oppositehypotenuse

BC = 16 × sin 42°

BC = 10.7 cm

Answer: 10.7 cm

2
Triangle ABC is right-angled at C. Angle A = 55° and AB = 24 cm. Find AC.
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cos 55° = adjacenthypotenuse

AC = 24 × cos 55°

AC = 13.8 cm

Answer: 13.8 cm

3
Triangle ABC is right-angled at C. Angle A = 27° and AB = 11 cm. Find BC.
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sin 27° = oppositehypotenuse

BC = 11 × sin 27°

BC = 4.99 cm

Answer: 4.99 cm

4
Triangle ABC is right-angled at C. Angle A = 63° and AB = 30 cm. Find AC.
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cos 63° = adjacenthypotenuse

AC = 30 × cos 63°

AC = 13.6 cm

Answer: 13.6 cm

5
Triangle ABC is right-angled at C. Angle A = 18° and AB = 9.50 cm. Find BC.
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sin 18° = oppositehypotenuse

BC = 9.50 × sin 18°

BC = 2.94 cm

Answer: 2.94 cm

Hard

Find the hypotenuse

Rearrange the ratio so the hypotenuse is the result.

Example
Triangle ABC is right-angled at C. Angle A = 35° and AC = 14 cm. Find the hypotenuse AB.

cos 35° = adjacenthypotenuse

cos 35° = 14AB

AB = 14cos 35°

AB = 17.1 cm

Answer: 17.1 cm

1
Triangle ABC is right-angled at C. Angle A = 40° and BC = 18 cm. Find AB.
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sin 40° = oppositehypotenuse

sin 40° = 18AB

AB = 18sin 40°

AB = 28.0 cm

Answer: 28.0 cm

2
Triangle ABC is right-angled at C. Angle A = 28° and AC = 9 cm. Find AB.
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cos 28° = adjacenthypotenuse

cos 28° = 9AB

AB = 9cos 28°

AB = 10.2 cm

Answer: 10.2 cm

3
Triangle ABC is right-angled at C. Angle A = 34° and AC = 22 cm. Find BC and AB.
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tan 34° = BC22

BC = 22 × tan 34°

BC = 14.8 cm

cos 34° = 22AB

AB = 22cos 34°

AB = 26.5 cm

Answer: BC = 14.8 cm, AB = 26.5 cm

4
Triangle ABC is right-angled at C. BC = 3 cm and AC = 40 cm. Find angle A.
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tan A = 340

A = 4.3°

Answer: 4.3°

5
Triangle PQR is right-angled at Q. PQ = 11 cm and PR = 15 cm. Find angle P.
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cos P = adjacenthypotenuse

cos P = 1115

P = 42.8°

Answer: 42.8°

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