Question

find the value of sin 15° by using the only value of sin 30°​

Answer 1

Answer:

For all values of the angle A we know that, (sin A2 + cos A2)2  = sin2 A2  + cos2 A2  + 2 sin A2 cos A2   = 1 + sin A  

Therefore, sin A2  + cos A2  = ± √(1 + sin A), [taking square root on both the sides]

Now, let A = 30° then, A2 = 30°2 = 15° and from the above equation we get,

sin 15° + cos 15° = ± √(1 + sin 30°)                       ….. (i)

Similarly, for all values of the angle A we know that, (sin A2 - cos A2)2  = sin2 A2 + cos2 A2 - 2 sin A2 cos A2  = 1 - sin A  

Therefore, sin A2  - cos A2  = ± √(1 - sin A), [taking square root on both the sides]

 

Now, let A = 30° then, A2 = 30°2 = 15° and from the above equation we get,

sin 15° - cos 15°= ± √(1 - sin 30°)                  …… (ii)

Clearly, sin 15° > 0 and cos 15˚ > 0

Therefore, sin 15° + cos 15° > 0

Therefore, from (i) we get,

sin 15° + cos 15° = √(1 + sin 30°)                                  ..... (iii)

Again, sin 15° - cos 15° = √2 (1√2 sin 15˚ - 1√2 cos 15˚)

or, sin 15° - cos 15° = √2 (cos 45° sin 15˚ - sin 45° cos 15°)

or, sin 15° - cos 15° = √2 sin (15˚ - 45˚)

or, sin 15° - cos 15° = √2 sin (- 30˚)

or, sin 15° - cos 15° = -√2 sin 30°

or, sin 15° - cos 15° = -√2 ∙ 12

or, sin 15° - cos 15° = - √22

Thus, sin 15° - cos 15° < 0

Therefore, from (ii) we get, sin 15° - cos 15°= -√(1 - sin 30°)    ..... (iv)

Now, adding (iii) and (iv) we get,

2 sin 15° = 1+12−−−−−√−1−12−−−−−√

2 sin 15° = 3√−1/2√

sin 15° = 3√−122√

Therefore, sin 15° = 3√−1/22√

Step-by-step explanation: