State the number of possible triangles that can be formed using the given measurements

Question

Hiroko Cabezas · Accepted Answer

Step 1
In trigonometry, the formula for the law of sines, or sine rule is the relation between the lengths of the sides of a triangle to the sines of its angles,

\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Step 2
Use the law of sines, to find the value of angle B, substitute angle C with

62^{\circ}

, side b with 9 and side c with 12. Multiply both sides by 9 to isolate sin B and further simplify to get the two values of

{41.45}^{\circ}

and

{138.55}^{\circ}

. Then, to find angle A, subtract the sum of angle B and angle C from

180^{\circ}

. For

∠ B = {41.45}^{\circ}

, the value of angle A is

{76.55}^{\circ}

and for

∠ B = {138.55}^{\circ}

, the value of angle A is

- {20.55}^{\circ}

and since

∠ A

cannot be negative. Hence, there is only one triangle possible.

\frac{\sin B}{b} = \frac{\sin C}{c}

\frac{\sin B}{9} = \frac{{\sin 62}^{\circ}}{12}

\sin B = \frac{3 (0.8829)}{4}

= 0.662

B = \sin^{- 1} (0.662)

= {41.45}^{\circ}

or

{138.55}^{\circ}

∠ A = 180^{\circ} - (∠ B + ∠ C)

= 180^{\circ} - ({41.45}^{\circ} + 62^{\circ})

= 180^{\circ} - {103.45}^{\circ}

= {76.55}^{\circ}

∠ A = 180^{\circ} - (∠ B + ∠ C)

= 180^{\circ} - ({138.55}^{\circ} + 62^{\circ})

= 180^{\circ} - {200.55}^{\circ}

= - {20.55}^{\circ}

State the number of possible triangles that can be formed

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