 # Value of sin 15

The whole trigonometric functions and formulas are designed based on three primary ratios. These ratios are Sine, cosine, and tangent in trigonometry. These ratios help us in finding angles and lengths of sides of a right triangle. We use sin, cos, and tan to calculate these angles. The basic degrees are 0, 30, 45, 60, 90 degrees and more. We use these degrees to find the value of the other trigonometric angles like the value of sine 15 degrees.

## Sin 15°= ?

 Sin 15 = (√3−1)/(2√2)

## How to find the Value of Sin 15 Degree?

• 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)

add (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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