Sine Cosine Rule And Area Of Triangles

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Apuntes de la lección

The Sine Rule

Used in non-right-angled triangles to find missing side lengths or angles.
States: a / sin A = b / sin B = c / sin C, where a is opposite A, etc.
To find a missing length, use two equal parts of the rule and solve.
To find a missing angle, rearrange to sin A / a = sin B / b = sin C / c.
Ambiguous case: given two sides and a non-included angle, there may be two possible triangles (acute and obtuse).
If the required angle is obtuse, use: obtuse angle = 180° – acute angle from calculator.

Triangle ABC

The Cosine Rule

Used in non-right-angled triangles when you have two sides and the included angle (to find the third side) or all three sides (to find an angle).
For a side: a² = b² + c² – 2bc cos A, where A is the angle between b and c.
For an angle: cos A = (b² + c² – a²) / (2bc).
No ambiguous case – the cosine rule gives a unique angle.
Label sides carefully: side a is opposite angle A.

Triangle ABC

Area of a Triangle

For any triangle: Area = ½ ab sin C, where C is the angle between sides a and b.
If C = 90°, sin 90° = 1, so Area = ½ × base × height (right-angled triangle).
Ensure all lengths are in the same units before calculating area.
If the included angle is not given, use sine or cosine rule first to find it.

Triangle ABC

Deciding the Trig Rule

Sine rule: use when you have an opposite pair (side and angle) and need another side or angle.
Cosine rule: use when you have two sides and the included angle (to find the third side) or all three sides (to find an angle).
Area rule: use when you have two sides and the included angle (to find area).
If no rule fits directly, use angles in a triangle sum to 180° to find a missing angle.
Harder questions may require multiple trig rules in sequence.

Deciding the Trig Rule

Worked Example – Sine Rule

Given triangle ABC with AB = 8.1 cm, BC = 12.3 cm, angle BCA = 27°.
Find angle x (at A): use sin x / 12.3 = sin 27° / 8.1 → x = sin⁻¹(12.3 sin 27° / 8.1) ≈ 43.6°.
Find side y (AC): first find angle ABC = 180° – 27° – 43.6° = 109.4°, then y / sin 109.4° = 8.1 / sin 27° → y ≈ 16.8 cm.

Triangle ABC

Worked Example – Cosine Rule

Given triangle ABC with AB = 4.2 km, BC = 3.8 km, AC = 7.1 km.
Find angle ABC: use cos θ = (4.2² + 3.8² – 7.1²) / (2 × 4.2 × 3.8) → θ = cos⁻¹(...) ≈ 125.0°.

Triangle ABC

Worked Example – Area of a Triangle

Given triangle ABC with AB = 32 cm, AC = 1.1 m, angle BAC = 74°.
Convert to same units: AB = 0.32 m, AC = 1.1 m.
Area = ½ × 1.1 × 0.32 × sin 74° ≈ 0.169 m² (3 s.f.).

Triangle ABC

Worked Example – Multiple Rules

Find area of triangle with sides 4.4 cm, 7.4 cm, 4.8 cm.
Use cosine rule to find an angle: cos ABC = (4.4² – 7.4² – 4.8²) / (–2 × 7.4 × 4.8) → ABC ≈ 34.65°.
Then area = ½ × 4.8 × 7.4 × sin 34.65° ≈ 10.1 cm² (3 s.f.).

Triangle ABC

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Preguntas de práctica

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1.What is the formula for the area of a triangle given two sides and the included angle?
Easy
- AArea = 1/2 ab sin C
- BArea = 1/2 ab cos C
- CArea = ab sin C
- DArea = 1/2 ab tan C
2.The sine rule states that for any triangle ABC:
Easy
- Aa/sin A = b/sin B = c/sin C
- Ba/sin B = b/sin C = c/sin A
- Csin A/a = sin B/b = sin C/c
- Da sin A = b sin B = c sin C
3.Which rule should be used to find a side when you know two sides and the included angle?
Easy
- ACosine rule
- BSine rule
- CArea rule
- DPythagoras' theorem
4.In the cosine rule a² = b² + c² - 2bc cos A, what does angle A represent?
Easy
- AThe angle opposite side a
- BThe angle between sides b and c
- CThe angle opposite side b
- DThe angle between sides a and c
5.Triangle ABC has AB = 8 cm, AC = 5 cm and angle BAC = 30°. Find the area of the triangle.
Easy
- A10 cm²
- B20 cm²
- C40 cm²
- D34.6 cm²
6.In triangle XYZ, XY = 7 cm, XZ = 9 cm and angle YXZ = 60°. Use the cosine rule to find YZ.
Medium
- A√(49 + 81 - 2×7×9×cos 60°) = √(130 - 63) = √67 ≈ 8.19 cm
- B√(49 + 81 - 2×7×9×sin 60°) = √(130 - 108.9) = √21.1 ≈ 4.59 cm
- C√(49 + 81 - 2×7×9×cos 30°) = √(130 - 108.9) = √21.1 ≈ 4.59 cm
- D√(49 + 81 - 2×7×9×tan 60°) = √(130 - 218.2) = negative
7.In triangle PQR, PQ = 10 cm, PR = 14 cm and angle PQR = 40°. Which rule can be used to find angle PRQ?
Easy
- ASine rule
- BCosine rule
- CArea rule
- DPythagoras' theorem
8.A triangle has sides of lengths 5 cm, 6 cm and 7 cm. Find the angle opposite the side of length 7 cm.
Medium
- A78.5°
- B44.4°
- C57.1°
- D101.5°

Más preguntas

9.Triangle ABC has AB = 6 cm, BC = 8 cm and angle ABC = 45°. Find AC.
Medium
- A√(36 + 64 - 2×6×8×cos 45°) = √(100 - 96×0.7071) = √(100 - 67.88) = √32.12 ≈ 5.67 cm
- B√(36 + 64 - 2×6×8×sin 45°) = √(100 - 96×0.7071) = √(100 - 67.88) = √32.12 ≈ 5.67 cm
- C√(36 + 64 - 2×6×8×tan 45°) = √(100 - 96×1) = √4 = 2 cm
10.In triangle ABC, AB = 8 cm, BC = 10 cm and angle ABC = 120°. Find the length of AC.
Medium
- A√(64 + 100 - 2×8×10×cos 120°) = √(164 - 160×(-0.5)) = √(164 + 80) = √244 ≈ 15.62 cm
- B√(64 + 100 - 2×8×10×cos 60°) = √(164 - 160×0.5) = √(164 - 80) = √84 ≈ 9.17 cm
- C√(64 + 100 - 2×8×10×sin 120°) = √(164 - 160×0.866) = √(164 - 138.56) = √25.44 ≈ 5.04 cm
- D√(64 + 100 - 2×8×10×cos 120°) = √(164 - 160×0.5) = √(164 - 80) = √84 ≈ 9.17 cm
11.Triangle PQR has sides PQ = 9 cm, PR = 12 cm and QR = 15 cm. Find the area of the triangle.
Medium
- A54 cm²
- B108 cm²
- C67.5 cm²
- D135 cm²
12.In triangle ABC, AB = 7 cm, AC = 9 cm and angle ABC = 50°. Find angle ACB.
Medium
- Asin⁻¹(7×sin 50°/9) ≈ 36.5°
- Bsin⁻¹(9×sin 50°/7) ≈ 80.0°
- Ccos⁻¹((7²+9²-?²)/(2×7×9))
- D180 - 50 - 36.5 = 93.5°
13.A triangle has sides of lengths 8 cm, 15 cm and 17 cm. Find the smallest angle.
Medium
- Acos⁻¹((15²+17²-8²)/(2×15×17)) = cos⁻¹(450/510) ≈ 28.1°
- Bcos⁻¹((8²+17²-15²)/(2×8×17)) = cos⁻¹(128/272) ≈ 61.9°
- Ccos⁻¹((8²+15²-17²)/(2×8×15)) = cos⁻¹(0/240) = 90°
- Dsin⁻¹(8/17) ≈ 28.1°
14.In triangle ABC, AB = 10 cm, BC = 14 cm and angle ABC = 30°. Find the area of the triangle.
Easy
- A35 cm²
- B70 cm²
- C60.6 cm²
- D121.2 cm²
15.Which of the following is NOT a correct form of the cosine rule?
Easy
- Aa² = b² + c² - 2bc cos A
- Bcos A = (b² + c² - a²)/(2bc)
- Ca² = b² + c² + 2bc cos A
- Db² = a² + c² - 2ac cos B
16.In triangle ABC, A = 40°, B = 60°, and side a = 8 cm. Find side b.
Medium
- A8×sin 60°/sin 40° ≈ 10.8 cm
- B8×sin 40°/sin 60° ≈ 5.9 cm
- C8×cos 60°/cos 40° ≈ 5.2 cm
- D8×tan 60°/tan 40° ≈ 16.5 cm
17.In triangle ABC, AB = 7 cm, BC = 9 cm and angle ABC = 50°. Which rule should be used to find AC?
Easy
- ASine rule
- BCosine rule
- CArea rule
- DPythagoras' theorem
18.What is the area of triangle PQR if PQ = 8 cm, PR = 6 cm and angle QPR = 30°?
Easy
- A12 cm²
- B24 cm²
- C48 cm²
- D20 cm²
19.In triangle ABC, AB = 10 cm, BC = 12 cm and angle ABC = 60°. Use the cosine rule to find AC.
Medium
- A11.1 cm
- B13.1 cm
- C15.6 cm
- D9.5 cm
20.In triangle XYZ, XY = 6 cm, XZ = 8 cm and angle YXZ = 40°. Find the area of the triangle.
Easy
- A15.4 cm²
- B24 cm²
- C19.2 cm²
- D30.8 cm²
21.Triangle ABC has sides AB = 9 cm, BC = 11 cm, CA = 7 cm. Find the angle opposite side BC.
Medium
- A85.9°
- B94.1°
- C50.5°
- D39.5°
22.Which of the following is the correct formula for the sine rule?
Easy
- Aa/sin A = b/sin B = c/sin C
- Bsin A/a = sin B/b = sin C/c
- Ca² = b² + c² - 2bc cos A
- DArea = ½ ab sin C
23.In triangle ABC, angle A = 40°, angle B = 60°, side a = 8 cm. Find side b using the sine rule.
Medium
- A10.8 cm
- B6.2 cm
- C12.3 cm
- D9.4 cm
24.Triangle PQR has PQ = 7 cm, QR = 9 cm, PR = 10 cm. Find the area of triangle PQR.
Hard
- A30.6 cm²
- B31.5 cm²
- C28.0 cm²
- D35.0 cm²
25.In triangle ABC, AB = 8 cm, BC = 14 cm, angle ABC = 30°. Find the area of triangle ABC.
Easy
- A28 cm²
- B56 cm²
- C48 cm²
- D32 cm²
26.In triangle ABC, AB = 6 cm, AC = 9 cm, angle BAC = 120°. Find BC.
Medium
- A13.1 cm
- B7.9 cm
- C15.0 cm
- D11.2 cm
27.Triangle ABC has sides AB = 5 cm, BC = 6 cm, CA = 7 cm. Find the smallest angle.
Medium
- A44.4°
- B57.1°
- C78.5°
28.Triangle BDC has BD = 11.6 cm, BC = 21.3 cm and angle BDC = 123.5°. Calculate angle BCD, correct to 1 decimal place.
Hard
- Asin⁻¹(11.6 × sin 123.5° / 21.3) ≈ 27.0°
- Bsin⁻¹(11.6 × sin 123.5° / 21.3) ≈ 33.5°
- C180° − 123.5° − 11.6° ≈ 44.9°
- Dsin⁻¹(21.3 × sin 123.5° / 11.6) — no solution (ratio exceeds 1)
29.In triangle ABC, AB = 7.4 cm, AC = 5.8 cm, angle BAC = 80°. Find the length of BC.
Medium
- A8.6 cm
- B6.2 cm
- C9.4 cm
- D7.0 cm
30.Triangle ABC has area 50 cm², AB = 10 cm, AC = 12 cm. Find the value of sin(angle BAC).
Medium
- A0.833
- B0.417
- C0.667
31.In triangle ABC, angle A = 30°, side a = 5 cm, side b = 8 cm. Use the sine rule to find angle B.
Medium
- A53.1°
- B126.9°
- C53.1° or 126.9°
- D30°
32.What is the formula for the area of a triangle given two sides a and b and the included angle C?
Easy
- AArea = 1/2 ab sin C
- BArea = ab sin C
- CArea = 1/2 ab cos C
- DArea = 1/2 a b
33.In the cosine rule a² = b² + c² - 2bc cos A, what does angle A represent?
Easy
- AThe angle opposite side a
- BThe angle between sides b and c
- CThe angle between sides a and b
- DThe angle between sides a and c
34.Triangle ABC has sides AB = 8 cm, AC = 5 cm and angle BAC = 30°. Find the area of the triangle.
Easy
- A10 cm²
- B20 cm²
- C40 cm²
- D17.3 cm²
35.The sine rule states that for any triangle ABC, which of the following is correct?
Easy
- A\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
- B\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
- Ca \sin A = b \sin B = c \sin C
- Da² = b² + c² - 2bc \cos A
36.Which of the following is the correct formula for the area of a triangle given two sides a and b and the included angle C?
Easy
- AArea = \frac{1}{2}ab \sin C
- BArea = ab \sin C
- CArea = \frac{1}{2}ab \cos C
- DArea = \frac{1}{2}ab
37.The cosine rule can be used to find a missing angle when all three sides are known.
Easy
True or false?
38.In triangle ABC, AB = 6 cm, AC = 9 cm, and angle BAC = 120°. Which rule should be used to find BC?
Easy
- ASine rule
- BCosine rule
- CArea rule
- DPythagoras' theorem
39.Match each given information to the correct trigonometric rule.
Medium
- Two sides and the included angle
- Two angles and a side opposite one angle
- All three sides
- Cosine rule
- Sine rule
- Cosine rule (to find an angle)
40.Triangle XYZ has XY = 7 cm, XZ = 9 cm, and angle YXZ = 60°. Using the cosine rule, the length YZ is:
Medium
- A\√{49 + 81 - 2 \times 7 \times 9 \times \cos 60°}
- B\√{49 + 81 - 2 \times 7 \times 9 \times \sin 60°}
- C\√{49 + 81 + 2 \times 7 \times 9 \times \cos 60°}
- D\√{49 + 81 - 7 \times 9 \times \cos 60°}
41.Which of the following statements about the sine rule are true? (Select all that apply)
Medium
- AIt can be used to find a missing side when two angles and a side are known.
- BIt can be used to find a missing angle when two sides and a non-included angle are known.
- CIt can be used to find the area of a triangle.
- DIt is only valid for right-angled triangles.
42.Arrange the steps to find the area of triangle ABC given sides a, b, and c (all three sides known).
Medium
- Use cosine rule to find an angle.
- Apply area formula \frac{1}{2}ab \sin C.
- Check that you have two sides and the included angle.
- Substitute the known values into the area formula.
43.In the cosine rule a² = b² + c² - 2bc \cos A, what does angle A represent?
Easy
- AThe angle opposite side a
- BThe angle between sides b and c
- CThe angle opposite side b
- DThe angle opposite side c
44.Match each set of given information to the correct trigonometric rule to use.
Medium
- Two sides and the included angle
- Two sides and an angle opposite one of them
- All three sides
- Two angles and a side
- Cosine rule to find the third side
- Sine rule to find the opposite angle
- Cosine rule to find an angle
- Sine rule to find a side
45.Complete the statement about the sine rule.
Easy
The sine rule is used in ____ triangles and states that \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.
46.In triangle ABC, angle A = 40°, angle B = 60°, and side a = 8 cm. Use the sine rule to find side b. Give your answer correct to 3 significant figures.
Medium
47.Triangle ABC has sides AB = 8 cm, BC = 10 cm, and CA = 12 cm. Find the area of the triangle correct to 3 significant figures.
Hard
48.In triangle ABC, AB = 7.4 cm, AC = 5.8 cm, and angle BAC = 80°. Find the length of BC correct to 3 significant figures.
Medium
49.Write down the formula for the area of a triangle given two sides a and b and the included angle C.
Easy
50.In triangle ABC, angle A = 40°, angle B = 60°, and side a = 8 cm. Use the sine rule to find side b. Give your answer correct to 3 significant figures.
Medium
- A10.8 cm
- B11.8 cm
- C12.8 cm
- D9.8 cm
51.Triangle ABC has sides AB = 8 cm, BC = 10 cm, and CA = 12 cm. Find the area of the triangle correct to 3 significant figures.
Medium
- A47.7 cm²
- B39.7 cm²
- C35.7 cm²
- D43.7 cm²
52.In triangle ABC, AB = 7.4 cm, AC = 5.8 cm, and angle BAC = 80°. Find the length of BC correct to 3 significant figures.
Medium
- A9.57 cm
- B7.57 cm
- C10.57 cm
- D8.57 cm

Apuntes de repaso

## Sine, Cosine Rule & Area of Triangles ### Sine Rule For any triangle with angles A, B, C and opposite sides a, b, c: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] - Use to find a side when you have an opposite pair (side & angle). - To find an angle, flip: \[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\] - **Ambiguous case**: Given two sides and a non-included angle, there may be two possible triangles. Your calculator gives the acute angle; if the required angle is obtuse, use: obtuse angle = 180° - acute angle. ### Cosine Rule For any triangle: \[a^2 = b^2 + c^2 - 2bc\cos A\] - Use to find a side when you know two sides and the included angle. - To find an angle when you know all three sides, rearrange: \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\] - No ambiguous case. ### Area of a Triangle For any triangle: \[\text{Area} = \frac{1}{2}ab\sin C\] - C is the angle between sides a and b. - If C = 90°, this becomes ½ × base × height. ### Deciding Which Rule to Use | Given | Want | Use | |-------|------|-----| | Two sides and an angle opposite one | Angle opposite other side | Sine rule | | Two angles and a side opposite one | Side opposite other angle | Sine rule | | Two sides and included angle | Third side | Cosine rule | | All three sides | Any angle | Cosine rule | | Two sides and included angle | Area | Area = ½ab sin C | - Often need multiple rules in one problem. Remember angles in a triangle sum to 180°. **Key tip**: Check the number of marks – more marks often require multiple steps.

Sine Rule: For any triangle, a/sin A = b/sin B = c/sin C, where a is opposite A, etc. Used to find missing sides or angles in non-right triangles.
Cosine Rule: For any triangle, a² = b² + c² - 2bc cos A, where a is opposite A. Used to find a side given two sides and included angle, or an angle given three sides.
Area of a Triangle (using sine): Area = ½ ab sin C, where C is the angle between sides a and b. Works for any triangle.
Ambiguous Case: When using sine rule with two sides and a non-included angle, there may be two possible triangles (acute and obtuse). The calculator gives the acute angle; if the diagram suggests an obtuse angle, subtract from 180°.

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