Find the value of sin 15 using sin 30.
We can find the value of sin 15 using the value of sin 30.
Answer: The value of sin 15 using sin 30 is (√3−1)/2√2
Let see, how we can find the value of sin 15° using sin 30°.
Explanation:
(sin P/2 + cos P/2)2 = sin2 P/2 + cos2 P/2 + 2 sin P/2 cos P/2 = 1 + sin P [Using sin 2A = 2 sin A cos A]
sin P/2 + cos P/2 = ± √ (1 + sin P)
(sin P/2 - cos P/2)2 = sin2 P/2 + cos2 P/2 - 2 sin P/2 cos P/2 = 1 - sin P [Using sin 2A = 2 sin A cos A]
sin P/2 - cos P/2 = ± √ (1 - sin P)
If P = 30°, then P/2 = 30°/2 = 15°
Substituting P/2 = 15° in the sin P/2 + cos P/2 = ± √ (1 + sin P) equation, we get
sin 15° + cos 15° = ±√ (1 + sin 30°) …(1)
Similarly, we have
sin 15° – cos 15° = ±√ (1 – sin 30°) …(2)
Since 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 get, from (2), sin 15° – cos 15°= -√(1 – sin 30°) … (4)
Adding equations (3) and (4) we get,
2 sin 15° = √(1 + ½) – √(1 – ½)
2 sin 15° = (√3−1)/√2
∴ sin 15° = (√3−1)/2√2
Thus, sin 15° = (√3−1)/2√2
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