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Trigonometry

Question 9 of 9

Sine and Cosine Rules

Question

Use the cosine rule to solve for the unknown sides and angles given \(b = 40\), \(c = 20\), \(\angle A = 70^\circ\)

Show solution Hide solution Fully worked — 9 steps
  1. Use the cosine rule to solve for \(a\)
    \[ a^2 = b^2 + c^2 - 2bc\cos A \]
  2. Substitute the values and solve for \(a\)
    \[ a^2 = 40^2 + 20^2 - 2(40)(20)\cos 70^\circ \]
  3. Round the answer to the nearest whole number
    \[ a = 38.11 \approx 38 \]
  4. Use the cosine rule to solve for \(\angle B\)
    \[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]
  5. Substitute and solve for \(\angle B\)
    \[ \cos B = \frac{(38.11)^2 + (20)^2 - (40)^2}{2(38.11)(20)} = 0.1656 \]
  6. Round the answer to the nearest whole number
    \[ \angle B = 80.47 \approx 80^\circ \]
  7. Use the cosine rule to solve for \(\angle A\)
    \[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
  8. Substitute and solve for \(\angle A\)
    \[ \cos A = \frac{40^2 + 20^2 - (38.11)^2}{2(40)(20)} = 0.3423 \]
  9. Round the answer to the nearest whole number
    \[ \angle A = 69.98 \approx 70^\circ \]

    That is, side a is 38, angle B is 80 degrees and angle A is 70 degrees.