Find the value of sin 15 using sin 30

We can find the value of Sin 150 with the help of sin 30 degrees.

(Sin P/2 + Cos P/2)2 = Sin2 P/2 + Cos2 P/2 +2Sin P/2Cos P/2

= 1 + sinP

Sin P/2 + Cos P/2 = ± √ (1 + sin P)

If P = 300 so P/2 = 30/2 =150

Putting this value in the above equation:

Sin 150 + Cos 150 = ±√ (1 + sin 30) …(1)

Also, (Sin P/2 – Cos P/2)2 = Sin2 P/2 + Cos2 P/2 – 2Sin P/2Cos P/2

= 1 – sinP

Sin P/2 – Cos P/2 = ± √ (1 – sin P)

Putting this value in the above equation:

Sin 150 – Cos 150 = ±√ (1 – sin 30) …(2)

As seen, sin 15° > 0 and cos 15˚ > 0

hence, sin 15° + cos 15° > 0

From (1) we will get,

sin 15° + cos 15° = √ (1 + sin 30°) …(3)

Also, 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 x 1/2

or, sin 15° – cos 15° = – √2/2

So, sin 15° – cos 15° < 0

Now we got, from (2) sin 15° – cos 15°= -√(1 – sin 30°) … (4)

Adding eq. (3) and (4) we get,

2 sin 15° = √(1 + ½) – √(1 – ½)

2 sin 15° = (√3−1)/√2

∴ sin 15° = (√3−1)/2√2

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