Question

How do you solve the right triangle given $A={{52}^{\circ }}$ and $B=8.4$ ?

Answer 1

Hint: In these types of questions, one is required to find all the measurements of the given triangle, which includes all the angles and the length of all the three sides of the given triangle. Basic trigonometry rules and properties can help in calculating the length of the sides and the angle can be calculated using the angle sum property.

Complete step-by-step solution:
Given a right-angled triangle $\Delta ABC$, whose one acute angle measures $\angle A={{52}^{\circ }}$ and one of the side lengths is $B=8.4$ .
We have to find the values of the rest of the angles and the lengths of the rest of the sides.
Let us assume that $\angle A$ and $\angle C$ are the acute angles of the right-angled triangle $\Delta ABC$ and $\angle B$ is the right angle.
Now, we know that the sum of all the angles in a triangle is ${{180}^{\circ }}$ by the angle sum property of a triangle, therefore, we can say that,
$\angle A+\angle B+\angle C={{180}^{\circ }}$
Now, substituting the values of $\angle A$ and $\angle B$ in the above equation we get,
$\Rightarrow {{52}^{\circ }}+{{90}^{\circ }}+\angle C={{180}^{\circ }}$
Solving the above equation for $\angle C$ we get the value of $\angle C$ as,
$\Rightarrow \angle C={{38}^{\circ }}$
Therefore, the values of the angles of the given right-angled triangle are as follows:
$\angle A={{52}^{\circ }}$ , $\angle B={{90}^{\circ }}$ and $\angle C={{38}^{\circ }}$
Now, we have to calculate the lengths of the sides of the given right-angled triangle $\Delta ABC$ . We have the measure of one of the sides of the triangle that is given as $B=8.4$ in the question.
Since, $B$ is the side opposite to the angle $\angle B$ , therefore, it must be the hypotenuse of the right-angled triangle $\Delta ABC$ . To find the lengths of the remaining two sides of the triangle, we will use the sine rule in the right-angled triangle $\Delta ABC$ .
The sine rule states that:
$\dfrac{A}{\sin \angle A}=\dfrac{B}{\sin \angle B}=\dfrac{C}{\sin \angle C}$
Substituting the values in the above equation we get,
 $\Rightarrow \dfrac{A}{\sin {{52}^{\circ }}}=\dfrac{8.4}{\sin {{90}^{\circ }}}=\dfrac{C}{\sin {{38}^{\circ }}}$
Now, we know that $\sin {{90}^{\circ }}=1$ , therefore, we get,
$\Rightarrow A=8.4\times \sin {{52}^{\circ }}$ And $C=8.4\times \sin {{38}^{\circ }}$
Solving the above expression by substituting the required values, we get,
$\Rightarrow A=6.619$ And $\Rightarrow C=5.171$
Therefore, we get the lengths of the sides of the right-angled triangle $\Delta ABC$ as, $A=6.6$ , $B=8.4$ and $C=5.1$

Note: While solving these questions and applying the sine or the cosine rule one may have to deal with various values of angles that may not be conventional. In such cases, the trigonometric ratios for these angles can be easily calculated by using a log table. Therefore, students must know how to use a log table as it may prove to be very helpful.