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Exact Value of sin 15°

How to find the exact value of sin 15° using the value of sin 30°?

Solution:

For all values of the angle A we know that, (sin $\frac{A}{2}$ + cos $\frac{A}{2}$)$^{2}$ = sin$^{2}$ $\frac{A}{2}$ + cos$^{2}$ $\frac{A}{2}$ + 2 sin $\frac{A}{2}$ cos $\frac{A}{2}$ = 1 + sin A

Therefore, sin $\frac{A}{2}$ + cos $\frac{A}{2}$ = ± √(1 + sin A), [taking square root on both the sides]

Now, let A = 30° then, $\frac{A}{2}$ = $\frac{30°}{2}$ = 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 $\frac{A}{2}$ - cos $\frac{A}{2}$)$^{2}$ = sin$^{2}$ $\frac{A}{2}$ + cos$^{2}$ $\frac{A}{2}$ - 2 sin $\frac{A}{2}$ cos $\frac{A}{2}$ = 1 - sin A

Therefore, sin $\frac{A}{2}$ - cos $\frac{A}{2}$ = ± √(1 - sin A), [taking square root on both the sides]

Now, let A = 30° then, $\frac{A}{2}$ = $\frac{30°}{2}$ = 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 ($\frac{1}{√2}$ sin 15˚ - $\frac{1}{√2}$ 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°