Question

10. In A ABC, prove that
1) sin(A + B)=sin C​

Answer 1

Answer:

Step-by-step explanation:

Given ,

ABC forms a triangle

To Prove :-

sin(A + B) = sinC

How To Prove :-

As they said that ABC forms a triangle we need to equate that sum of all angles in a triangle to 180° and we need to find value of '(A + B)' in terms of 'C' and by applying 'sin' ratio on both sides we can prove that 'sin(A + B) = sinC'.

Formula Required :-

Sum of all angles in a triangle = 180°

In second quadrant 'sin' ratio is positive

→ sin(180° - C) = sinC

Solution :-

A + B + C = 180°

Transposing to 'C' to R.H.S :-

A + B = 180° - C

Applying 'sin' ratio on both sides :-

sin(A + B) = sin(180° - C)

                = sinC

[ ∴ 180° - C is in second quadrant and in second quadrant 'sin' ratio is positive ]

∴ sin(A + B) = sinC

Hence Proved.

Know More :-

In 1st quadrant :-

'sin' ratio is positive

'cos' ratio is positive

'tan' ratio is positive

In 2nd quadrant :-

'sin' ratio is positive

'cos' ratio is negative

'tan' ratio is negative

In 3rd quadrant :-

'sin' ratio is negative

'cos' ratio is negative

'tan' ratio is positive

In 4th quadrant :-

'sin' ratio is negative

'cos' ratio is positive

'tan' ratio is negative