Trigonometry: sine & cosine rule

For triangles without a right angle, the sine rule and cosine rule let you find unknown sides and angles. The sine rule states a/sin(A) = b/sin(B) = c/sin(C), useful when you know an angle and its opposite side. The cosine rule, c² = a² + b² − 2ab·cos(C), is useful when you know two sides and the angle between them, or all three sides.

Key terms

Sine rule:

a/sin(A) = b/sin(B) = c/sin(C), relating sides and their opposite angles.

Cosine rule:

c² = a² + b² − 2ab·cos(C), relating all three sides and one angle.

Questions

1. 1. The sine rule and cosine rule are used for:

   • Triangles without a right angle
   • Only right-angled triangles
   • Circles only
   • Squares only

2. 2. The sine rule states:

   • a/sin(A) = b/sin(B) = c/sin(C)
   • a + b = c always
   • a × b = c always
   • sin(A) = sin(B) = sin(C) always

3. 3. The cosine rule is most useful when you know:

   • Two sides and the angle between them
   • Only one single side
   • Nothing about the triangle
   • Only the triangle's colour

4. 4. The cosine rule formula is:

   • c² = a² + b² − 2ab·cos(C)
   • c = a + b
   • c² = a² + b² (always, for any triangle)
   • c = a × b

5. 5. The sine rule is especially useful when you know:

   • An angle and its opposite side
   • Nothing at all about the triangle
   • Only the triangle's perimeter
   • Only the triangle's area

6. 6. Pythagoras' theorem (a² + b² = c²) applies to:

   • Right-angled triangles specifically
   • Every triangle, with no exceptions
   • Only triangles with no right angle
   • Circles only

7. 7. A triangle with all three angles known but no sides:

   • Cannot have its exact side lengths found without at least one side length
   • Can always have its side lengths found from angles alone
   • Is impossible to exist
   • Must be a right-angled triangle

8. 8. A triangle has angle A = 40°, angle B = 60°, and side a = 10. Using the sine rule, side b can be found using:

   • 10/sin(40°) = b/sin(60°)
   • 10 + 40 = b + 60
   • 10 × 40 = b × 60
   • b = 10 − 40 + 60

9. 9. A triangle has sides a = 5, b = 7, and angle C = 60° between them. The cosine rule would find:

   • Side c
   • Angle A only, with no other use
   • Nothing useful
   • Only the perimeter

10. 10. Which rule would you use to find a missing angle if you know all three side lengths?

    • The cosine rule (rearranged)
    • The sine rule only
    • Pythagoras' theorem only
    • None of these can be used

11. 11. A triangular block of land has two angles of 65° and 45°, with the side between them measuring 50m. To find the other two sides, you would use:

    • The sine rule
    • Only Pythagoras' theorem
    • The area formula only
    • None of these

12. 12. The angles in any triangle (right-angled or not) always add up to:

    • 180°
    • 360°
    • 90°
    • 270°

13. 13. A triangle has sides a = 8, b = 6, and included angle C = 70°. Using the cosine rule, side c is found by first calculating:

    • 8² + 6² − 2(8)(6)cos(70°)
    • 8 + 6 − 2(8)(6)
    • 8² + 6² (Pythagoras only)
    • 8 × 6 × cos(70°)

14. 14. A triangle has angle A = 50°, side a = 12, and side b = 9. The sine rule can be used to find:

    • Angle B
    • Only the perimeter, with no angle found
    • Nothing useful from this information
    • The area only, with no angle found

15. 15. Why can't Pythagoras' theorem alone be used to solve a triangle with no right angle?

    • Pythagoras' theorem specifically relies on the 90° angle relationship, which non-right triangles don't have
    • Pythagoras' theorem works identically for every possible triangle
    • Non-right triangles have no sides to measure
    • There is no real difference between right and non-right triangles

16. 16. A ship sails on a bearing, and two lighthouses take angle readings to it from known positions and a known baseline distance. This scenario suits:

    • The sine rule, to find distances without direct measurement
    • Only Pythagoras' theorem
    • Simple addition of angles with no trigonometry
    • None of these methods

17. 17. A triangular garden has sides 12m, 15m, and 18m. To find one of its angles, you would use:

    • The cosine rule (rearranged to solve for an angle)
    • The sine rule only, with no other information
    • Pythagoras' theorem, since it works for any triangle
    • It is impossible to find any angle

18. 18. Why might surveyors and navigators rely heavily on the sine and cosine rules in real-world work?

    • They allow accurate distance and angle calculations without needing to physically measure every side directly
    • These rules have no practical, real-world application
    • Surveyors only ever work with right-angled triangles
    • Direct measurement is always easier than using these rules

19. 19. The "ambiguous case" of the sine rule refers to a situation where:

    • Given information can produce two possible valid triangles
    • The sine rule always gives one guaranteed unique answer
    • No triangle can ever be formed
    • The cosine rule must always be used instead

20. 20. A triangle has two sides and a non-included angle known. Which rule is most directly applicable here?

    • The sine rule
    • The cosine rule only
    • Neither rule applies to this situation
    • Only Pythagoras' theorem

21. 21. Why is understanding both the sine rule and cosine rule useful, rather than relying on just one?

    • Different given information (sides/angles known) suits different rules for solving a triangle efficiently
    • Only one of these two rules is ever actually useful
    • Both rules always give completely identical results for any triangle
    • These rules have no real distinction between them