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Trigonometry

Question 6 of 9

Area of Triangle

Question

In triangle ABC, \(b = 3\,\text{cm}\), \(a = 8\,\text{cm}\) and the area of the triangle is \(12\,\text{cm}^2\). Find the size of angle \(A\) (Round your answer to 2 decimal places)

Show solution Hide solution Fully worked — 5 steps
Triangle ABC, drawn to scale, with vertex A at the top left, C at the lower left and B at the right. The given sides are b = 3 from A to C and a = 8 from C to B; the third side, c, from A to B, is unlabelled. Angle A, at the top vertex, is marked with an arc — this is the angle the solution finds. The angle at C, which the solution shows to be a right angle, carries no mark in the figure.
  1. State the formula
    \[ \mathit{area} = \frac{1}{2}\,ab \sin C \]
  2. Substitute values based on formula
    \[ 12\,\text{cm}^2 = \frac{1}{2}(8\,\text{cm})(3\,\text{cm})(\sin C) \]
  3. Solve for angle C

    Evaluate \(\dfrac{1}{2}(8\,\text{cm})(3\,\text{cm})(\sin C)\):

    \[ 12\,\text{cm}^2 = (12\,\text{cm}^2)(\sin C) \]

    Divide both sides by \(12\,\text{cm}^2\):

    \[ \sin C = \frac{12\,\text{cm}^2}{12\,\text{cm}^2} = 1 \]

    Compute for \(\sin^{-1} 1\) to get \(\angle C\):

    \[ C = 90^\circ \]
  4. Since it's a right triangle, use trigonometry ratios
    \[ \tan A = \frac{8}{3} \]

    Divide 8 by 3:

    \[ \tan A = 2.67 \]
  5. Final answer

    Compute for \(\tan^{-1} 2.67\) to get angle A:

    \[ A = 69.44^\circ \]

    That is, the size of angle A is 69.44 degrees, to 2 decimal places.