How do you use Law of sines, given A=102, b=13, c=10?

Question

7360 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-26T04:17:05

$color(maroon)(a = 17.97, hat B = 45.03^@, hat C = 32.97^@$

$color(brown)("Area of the triangle " A_t = (1/2) b c sin A = color(brown)(63.58 " sq units"$

$hat A = 102^@, b = 13, c = 10$

We cannot directly use Law of Sines.

$"First let's use Law of Cosines to find the third side " color(brown)(a)$

![http://www.dummies.com/education/math/trigonometry/laws-of-sines-and-cosines/]( useruploads.socratic.org )

$a^2 = b^2 + c^2 - (2 b c cos A)$

$a = sqrt(13^2 + 10^2 - (2 * 13 * 10 * cos 102)) = 17.97 " units"$

Only one triangle is possible, having $" " hatA = 102^@$ an obtuse angle.

Since we know all the three sides and one angle, we can apply law of sines to find the other two angles.

$sin A / a = sin B / b = sin C / c$

$hat B = sin ^-1 ((b.sin A) / a) = sin^-1 ((13 * sin 102)/17.97) = 45.03^@$

Similarly, $hat C = sin _1 ((10 * sin 102) / 17.97) = 32.97^@$

$"Area of the triangle " A_t = (1/2) b c sin A = (1/2) * 13 * 10 * sin 102 = color(maroon)(63.58 " sq units"$