Question

How do you solve the triangle \[ABC\], given \[a = 10\], \[A = 35\], \[B = 25\]?

Answer 1

Hint: Here we need to solve the given triangle. First, we will find the value of the third angle using the property of the triangle, and then we will use the sine law of the triangle so as to get the value of the rest sides of the triangle as the value of only one side is given in the question. Using the sine law, we will get the values of the sides and hence this will be the required solution of the given triangle.

Formula used:
According to sine law \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\], where, \[a\], \[b\] and \[c\] are the sides of the triangle and \[A\] , \[B\] and \[C\] are the angles of the triangle.

Complete step by step solution:
Here we need to solve the given triangle \[ABC\]
It is given that:
\[a = 10\], \[A = 35\], \[B = 25\]
We will find the value of the third angle using one of the properties of the triangle according to which the sum of all the sides of a triangle is equal to \[180^\circ \].
\[ \Rightarrow \angle A + \angle B + \angle C = 180^\circ \]
Now, we will substitute the value of the given angles here.
\[ \Rightarrow 35^\circ + 25^\circ + \angle C = 180^\circ \]
On adding the angles, we get
\[ \Rightarrow 60^\circ + \angle C = 180^\circ \]
Now, we will subtract \[60^\circ \] from both sides.
\[ \Rightarrow 60^\circ + \angle C - 55^\circ = 180^\circ - 60^\circ \]
On further simplification, we get
\[ \Rightarrow \angle C = 120^\circ \]
Now, we will use the sine law of triangle according to which
\[ \Rightarrow \dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Here,
\[a\], \[b\] and \[c\] are the sides of the triangle and \[A\] , \[B\] and \[C\] are the angles of the triangle.
Now, we will substitute the value of sides and the angles of the triangle here.
\[ \Rightarrow \dfrac{{10}}{{\sin 35^\circ }} = \dfrac{b}{{\sin 25^\circ }} = \dfrac{c}{{\sin 120^\circ }}\]
Now, we will consider the first two ratios.
\[ \Rightarrow \dfrac{b}{{\sin 25^\circ }} = \dfrac{c}{{\sin 120^\circ }}\]
Now, we will use the sine table to get the value of \[\sin 35^\circ \] and \[\sin 25^\circ \]
\[\begin{array}{l}\sin 35^\circ = 0.5735\\\sin 25^\circ = 0.4226\end{array}\]
Now, we will substitute these values here.
\[ \Rightarrow \dfrac{{10}}{{0.5735}} = \dfrac{b}{{0.4226}}\]
Now, we will multiply the 0.4226 on both sides.
\[\begin{array}{l} \Rightarrow \dfrac{{10}}{{0.5735}} \times 0.4226 = \dfrac{b}{{0.4226}} \times 0.4226\\ \Rightarrow 7.3681 = b\\ \Rightarrow b = 7.3681\end{array}\]
Now, we will consider the last two ratios.
\[ \Rightarrow \dfrac{b}{{\sin 25^\circ }} = \dfrac{c}{{\sin 120^\circ }}\]
We know that the \[b = 7.3681\]
Now, we will use the sine table to get the value of \[\sin 120^\circ \] and \[\sin 25^\circ \]
\[\begin{array}{l}\sin 120^\circ = 0.8660\\\sin 25^\circ = 0.4226\end{array}\]
Now, we will substitute these values here.
\[ \Rightarrow \dfrac{{7.3681}}{{0.4226}} = \dfrac{c}{{0.8661}}\]
Now, we will multiply the 0.8661 on both sides.
\[\begin{array}{l} \Rightarrow \Rightarrow \dfrac{{7.3681}}{{0.4226}} \times 0.8661 =

\dfrac{c}{{0.8661}} \times 0.8661\\ \Rightarrow 15.0987 = c\\ \Rightarrow c = 15.0987\end{array}\]

Hence, these are the required angles and the sides of the triangle.

Note:
Here we have obtained the value of the sides and the value of all the angles using the property of the triangle and the sine law of triangles. We need to remember the basic properties of the triangle and some important laws to find the values of the sides and the angles of the triangle easily. Remember that the value of the sum of any two sides of the triangle is always greater than the third side of that triangle.