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Cambridge IGCSE0580

Non-right-angled triangles

Mathematics 0580 Chapter Notes

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Non-right-angled triangles
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In this chapter

  1. 1The sine rule
  2. 2The cosine rule for a side
  3. 3The cosine rule for an angle
  4. 4Obtuse angles and the ambiguous case
  5. 5Area of a triangle

1. The sine rule

Use the sine rule in any triangle where you know a side and the angle opposite it, plus one more side or angle. Side a is opposite angle A.

To find a length, put the unknown on top. To find an angle, put the sine on top. Triangles that are not right-angled have no hypotenuse, so sin, cos and tan of the right-angled kind do not apply.

a/(sin A) = b/(sin B) = c/(sin C)

(sin A)/a = (sin B)/b

Examiner insight

“As would be expected common incorrect approaches included attempting to use the cosine rule and working with sine, cosine or tangent having not recognised that these could not be applied as the triangle was not right angled.”

Worked example 14 marks

In triangle ABC, BC = 15 cm, AC = 15√3 cm and angle ABC = 120°. Show that triangle ABC is an isosceles triangle.

  1. 1

    BC is opposite A and AC is opposite B: (sin A)/15 = (sin 120°)/(15√3) [1]

  2. 2

    sin A = (15 × sin 120°)/(15√3) = (√3)/(2√3) = 1/2 [1]

  3. 3

    angle BAC = 30° [1]

  4. 4

    180° - 120° - 30° = 30°, so angle BCA = angle BAC: two equal angles, so the triangle is isosceles [1]

2. The cosine rule for a side

Use the cosine rule to find a side when you know the other two sides and the angle between them.

Write the rule, substitute the values, then work out. Take the square root last.

a² = b² + c² - 2bccos A

Examiner insight

“Candidates should be aware that no marks can be gained for quoting the cosine rule until relevant values have been substituted in.” “Premature rounding, both at the first stage and with some values in the cosine rule often resulted in final answers that were outside of the acceptable range.”

Worked example 13 marks

In triangle PQR, PQ = 8 cm, PR = 11 cm and angle QPR = 52°. Calculate QR.

  1. 1

    QR² = 8² + 11² - 2 × 8 × 11 × cos 52° [1]

  2. 2

    QR² = 185 - 176cos 52° = 76.64… [1]

  3. 3

    QR = √(76.64…) = 8.75 cm [1]

3. The cosine rule for an angle

When you know all three sides, use the cosine rule rearranged to make the angle the subject.

The largest angle is opposite the longest side. A negative value of cos A means the angle is obtuse, and the calculator gives it directly.

cos A = (b² + c² - a²)/2bc

Examiner insight

“Some candidates did not appreciate that the largest angle is opposite the longest side and so often all three angles were calculated.”

Worked example 13 marks

The sides of a triangle are 5 cm, 7 cm and 11 cm. Find the size of the largest angle in this triangle.

  1. 1

    largest angle x is opposite the 11 cm side: cos x = (5² + 7² - 11²)/(2 × 5 × 7) [1]

  2. 2

    cos x = (25 + 49 - 121)/70 = -47/70 [1]

  3. 3

    x = 132.2° [1]

4. Obtuse angles and the ambiguous case

The sine of an obtuse angle is positive, and sin(180° -x) = sin x. The calculator's sin⁻¹ always gives the acute angle. If the angle is obtuse, subtract it from 180°.

The ambiguous case can occur when you know two sides and an angle that is not between them. The sine rule then gives x and 180° - x. Keep the second only if the angles still total less than 180°.

sin(180° - x) = sin x

Examiner insight

“They were told that the angle at ABC was obtuse but they did not know to use 180 – 35.5.” “Some candidates used the cosine rule to set up a quadratic equation in x for the missing side of the triangle, solved this correctly and chose the appropriate value for the side length to use in the sine rule, but then overlooked the information that y was obtuse and so gave the calculator value of 63.16 for y.”

Worked example 14 marks

In triangle ABC, AB = 15 cm, BC = 12 cm and angle BAC = 40°. Angle ACB is obtuse. Calculate angle ACB.

  1. 1

    AB is opposite C and BC is opposite A: (sin C)/15 = (sin 40°)/12 [1]

  2. 2

    sin C = (15 × sin 40°)/12 = 0.8035… [1]

  3. 3

    calculator gives the acute angle, 53.5° [1]

  4. 4

    C is obtuse, so C = 180° - 53.5° = 126.5° [1]

5. Area of a triangle

Area of a triangle is half the product of two sides and the sine of the angle between them. Use it when you do not know the height. The formula is given in the List of formulas, with the sine and cosine rules.

The half must be there. The angle must be the one included between the two sides. It works for an obtuse angle too, because its sine is positive.

area = 1/2absin C

Examiner insight

“Many of those who made some progress neglected the half in the formula for area of a triangle.” “Some candidates did not score because they used cos rather than sin in the formula.”

Worked example 12 marks

CX = 5 cm and XD = 8 cm. Angle CXD = 80.5°. Work out the area of triangle CXD.

  1. 1

    area = 1/2 × 5 × 8 × sin 80.5° [1]

  2. 2

    = 19.7 cm² [1]

Recap

  • Sine rule: each side over the sine of its opposite angle. Needs a side and its opposite angle.
  • Cosine rule for a side: a² = b² + c² - 2bccos A. Two sides and the angle between them.
  • Cosine rule for an angle: cos A = (b² + c² - a²)/2bc. Three sides. Largest angle is opposite the longest side.
  • The calculator's sin⁻¹ gives the acute angle. For an obtuse one, use 180° - x.
  • Ambiguous case: two sides and a non-included angle can give two triangles.
  • Area = 1/2absin C, with C between sides a and b.
  • Never round in the middle of a calculation. Substitute before working out.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. In triangle ABC, angle A = 30°, angle B = 45° and BC = 8 cm. Find AC. Give your answer as a surd.3 marks
  2. Do not use a calculator. A triangle has sides of 10 cm and 12 cm. The angle between them is 30°. Work out the area of the triangle.2 marks
  3. Do not use a calculator. In triangle PQR, PQ = 3 cm, PR = 8 cm and angle QPR = 60°. Find QR.3 marks
  4. Do not use a calculator. The sides of a triangle are 3 cm, 5 cm and 7 cm. Find the size of the largest angle.3 marks
  5. In triangle LMN, LM = 12.4 cm, MN = 9.8 cm and angle LNM = 71°. Calculate angle NLM.3 marks
  6. A triangle has sides of 6.4 cm and 9.5 cm. The angle between them is 125°. Calculate the area of the triangle.2 marks
  7. In triangle XYZ, XY = 14 cm, XZ = 9 cm and angle YXZ = 115°. Calculate YZ.3 marks
  8. In triangle ABC, angle BAC = 40°, AB = 10 cm and BC = 7 cm. Angle ACB is not known to be acute or obtuse. Calculate the two possible sizes of angle ACB.4 marks
  9. Do not use a calculator. A triangular flower bed has two edges of 6 m and 10 m. The angle between these two edges is obtuse. The area of the flower bed is 15√3 m². Work out the perimeter of the flower bed.5 marks
  10. Four marker posts A, B, C and D stand on level ground. Triangle ABC and triangle ACD share the side AC. AB = 75 m, BC = 52 m and angle BAC = 38°. Angle ACB is obtuse. AD = 60 m and CD = 40 m. Calculate angle ADC.7 marks

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