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Trigonometry

Question 8 of 9

Sine and Cosine Rules

Question

Use the cosine rule to solve for the unknown sides and angles

Triangle ABC, drawn to scale from the given values. The base AB, of length 15 (side c), runs from A on the left to B on the right; the side of length 10 (side a) runs from B up to C; and the angle at B, marked by the arc, is 28°. The side from A to C is labelled b = ? — the side to be found. The unmarked arcs at A and C are the angles to be found; the angle at C is obtuse.
Show solution Hide solution Fully worked — 10 steps
  1. Use the cosine rule to solve for \(b\)
    \[ b^2 = a^2 + c^2 - 2ac\cos B \]
  2. Substitute the values and solve for \(b\)
    \[ b^2 = 10^2 + 15^2 - 2(10)(15)\cos 28^\circ \]
  3. Get the square root of both sides
    \[ b^2 = 60.1157 \]
  4. Round the answer to the nearest whole number
    \[ b = 7.75 \approx 8 \]
  5. Use the cosine rule to solve for \(\angle A\)
    \[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
  6. Substitute values and solve for \(\angle A\)
    \[ \cos A = \frac{(7.75)^2 + 15^2 - 10^2}{2(7.75)(15)} \]
  7. Round the answer to the nearest whole number
    \[ \angle A = 37.25^\circ \approx 37^\circ \]
  8. Use the cosine rule to solve for \(\angle C\)
    \[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
  9. Substitute the values and solve for \(\angle C\)
    \[ \cos C = \frac{10^2 + (7.75)^2 - 15^2}{2(10)(7.75)} \]
  10. Round the answer to the nearest whole number
    \[ \angle C = 114.77 \approx 115^\circ \]

    That is, side b is 8, angle A is 37 degrees and angle C is 115 degrees.