Question

In triangle ABC, A =$30^{\circ}$ , b = 8, a = 6, then $B = sin^{-1}x$, where x =
A. $\dfrac{1}{2}$
B. $\dfrac{1}{3}$
C. $\dfrac{2}{3}$
D. 1

Answer 1

Hint: We have a triangle ABC with A = $30^{\circ}$, b = 8 and a = 6. We are given $B = sin^{-1}x$, we need to find the value for B, that is we need to find the value for $sin B$. The law of sines.
Formula used:
$\dfrac{sin A}{a} = \dfrac{sin B}{b} = \dfrac{sin C}{c}$
Complete Step-by-Step : We We have a triangle ABC with A = $30^{\circ}$, b = 8 and a = 6.
We have $B = sin^{-1}x$
$\implies x = sin B$
Therefore, to find the value of x, it is enough to calculate sin B.
By the law of sines we have,
$\dfrac{sin A}{a} = \dfrac{sin B}{b} = \dfrac{sin C}{c}$
Substituting the values for a, A and b in the above ratio we get,
$\dfrac{sin 30^{\circ}}{6}=\dfrac{sin B}{8}$
$sin B = \dfrac{sin 30^{\circ}\times 8}{6}$
$sin B = \dfrac{\dfrac{1}{2}\times 8}{6}$
$sin B = \dfrac{8}{6\times 2}$
$\implies sin B =\dfrac{2}{3}$
Therefore, the value of x is $\dfrac{2}{3}$. So, the answer is Option C.
Note: Remember whenever we need to find an unknown angle or an unknown side for a triangle we use the trigonometric ratios if the given triangle is a right-angle triangle. If the given triangle is not a right-angle then always use the law of sines (sine rule) or the law of cosines (cosine rule) to find an unknown angle or an unknown side.