Question

In a triangle$\vartriangle ABC$ , if $b=20,c=21$, and $\sin A=\frac{3}{5}$ then $a=$
A. $12$
B. $13$
C. $14$
D. $15$

Answer 1

Hint: _{We will} convert the angle $\sin A\,$_{to$\,\cos A$} with the help of Pythagoras theorem and formula of the trigonometric ratios. Then we will use cosine rule or Law of cosine ${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$ and substitute all the values to find the length of the side $a$of the triangle $\vartriangle ABC$_.
 Formula used:
Pythagoras Theorem: ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$
Trigonometric ratios:
$\begin{align}
  & \sin A=\frac{P}{H} \\
 & \cos A=\frac{B}{H} \\
\end{align}$
Here $P$ is perpendicular, $B$ is base and $H$ is hypotenuse of a right angled triangle.
Complete step-by-step solution:
We are given a triangle _having values of the length of the sides $b=20,c=21$_and angle $\sin A=\frac{3}{5}$_and we have to find the value of the side $a$.
We will first convert the angle $\sin A\,$_{to$\,\cos A$}.
As given $\sin A=\frac{3}{5}$ _{and we} know $\sin A=\frac{P}{H}$ so comparing both we will get,
$\begin{align}
  & P=3 \\
 & H=5 \\
\end{align}$
Now we will use Pythagoras theorem and find the value of the base $B$.
$\begin{align}
  & {{5}^{2}}={{3}^{2}}+{{B}^{2}} \\
 & 25=9+{{B}^{2}} \\
 & {{B}^{2}}=16 \\
 & B=4
\end{align}$
So the value of the angle $\,\cos A$ will be,
$\,\cos A=\frac{4}{5}$
We will now substitute all the values in the cosine rule to find the value of the length of the side $a$,
$\begin{align}
  & {{a}^{2}}={{(20)}^{2}}+{{(21)}^{2}}-2(20)(21)\frac{4}{5} \\
 & {{a}^{2}}=400+441-672 \\
 & {{a}^{2}}=169 \\
 & a=13 \\
\end{align}$
The triangle $\vartriangle ABC$ having sides $b=20,c=21$, and angle $\sin A=\frac{3}{5}$_{, the value of the length of the side $a$} of the triangle is $a=13$.Hence the correct option is (B).
Note:
Trigonometric ratios are the ratios of the length of the sides of a right angled triangle with respect to the angle. There are six trigonometric functions that are sine, cosine, tangent, cotangent, secant, and cosecant.
We have used Pythagoras theorem in the solutions because it states the relationship between the three sides perpendicular, hypotenuse and base of the right angled triangle.