Question

In a triangle \[ABC\], \[a = 7\], \[A = {30^ \circ }\], \[B = {70^ \circ }\]. What is angle \[C\] and length of side \[c\]?

Answer 1

Hint: To solve this question, we will first draw a rough figure and specify the given data in that figure. Then we will apply the angle sum property of a triangle, which states that the sum of all three angles of a triangle is \[{180^ \circ }\], to find the angle \[c\]. Then we will use the sine theorem to find the length of the side \[c\].
Formula used:
Sine theorem:- \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Here, \[a,b,c\] are the sides opposite to angles \[A,B,C\] respectively.

Complete step-by-step solution:
First, we will draw a rough figure with the data given in the question.

Now we will find \[\angle C\] using the angle sum property of the triangle.
According to the angle sum property of a triangle;
\[\angle A + \angle B + \angle C = {180^ \circ }\]
On shifting we get;
\[ \Rightarrow \angle C = {180^ \circ } - \angle A - \angle B\]
Putting the value of \[\angle A,\angle B\] we get;
\[ \Rightarrow \angle C = {180^ \circ } - {30^ \circ } - {70^ \circ }\]
So, we get;
\[ \Rightarrow \angle C = {80^ \circ }\]
Now to find the side \[c\], we will use the sine theorem. So, according to the sine theorem we have; \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Now since we have to find side \[c\] and we know the value of side \[a\], so we use;
\[ \Rightarrow \dfrac{a}{{\sin A}} = \dfrac{c}{{\sin C}}\]
On cross multiplication we get;
\[ \Rightarrow c = \dfrac{{a\sin C}}{{\sin A}}\]
Now we will put the values. So, we get
\[ \Rightarrow c = \dfrac{{7\sin {{80}^ \circ }}}{{\sin {{30}^ \circ }}}\]
We know, \[\sin {30^ \circ } = \dfrac{1}{2}\] and we can find \[\sin {80^ \circ }\] from the calculator, \[\sin {80^ \circ } = 0.98480\].
\[ \Rightarrow c = \dfrac{{7\left( {0.9848} \right)}}{{\left( {\dfrac{1}{2}} \right)}}\]
On simplification we get;
\[ \Rightarrow c = 7 \times 2 \times \left( {0.9848} \right)\]
\[ \Rightarrow c = 13.7872\]

Note: One important thing we can note here is that the angles are represented by the capital letters and the sides are represented by the small letters. Also, we can note that the sides are marked opposite to their angles. For example, in the above figure side \[a\] is marked opposite to \[\angle A\] and similarly, the other sides are also marked in the same way.