Question

How do I find the missing angles of a triangle?
(a) Use the fact that the sum of the angles of the triangle is 180 degree
(b) Use sine formula $\left[ \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C} \right]$
(c) Use cosine formula $\left[ \cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc} \right]$
(d) All of the above

Answer 1

Hint: We are to find the ways to find the angles of a triangle in this problem. So, we will start with the given options and then analyze each of them to get if they are helping us to find the angles of the triangles and then choose the right ones. One or more options can be true for these kinds of questions.
According to the question, we have to find the way to find the missing angles of a triangle and we are given some of the options. We are to analyze the options and find the right solution to get through our given problem.

Complete step by step solution:
So, to start with, as we all know, from the given first option, the sum of all the angles of a triangle is always 180 degrees. Hence, we can conclude that the first option is true.
Coming to the second option, we are given that we can use the sine formula of triangles to find the angles of the triangle, which is said to be, $\left[ \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C} \right]$, where A,B,C respectively denoting angles A,B and C and a,b and c denoting the sides opposite to angle A,B and C respectively.
From this, if we are given any two sides and one angle or any two angles of one side of a triangle we can find the other remaining angles and sides as per our own need.
Hence, this is also a correct option.
Now, if we check the third option, we can see that the cosine formula of triangles is used here, which is said to be, $\left[ \cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc} \right]$, where A,B,C respectively denoting angles A,B and C and a,b and c denoting the sides opposite to angle A,B and C respectively.
So, if the sides of any triangle are given we can find all the angles of the triangle from this formula. Hence, this is also a right option.
Hence the solution is, (d) All of the above.

Note: Here in this problem we have used the trigonometric angles identities to execute the solutions and get the right options. For the sine formula, it can be proved with the help of another important property, the angle subtended at the centre by any chord to a circle is twice the angle subtended by it at any point on the circumference. And similarly the cosine formula can be proved with the help of the vector algebra very easily.