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1800-102-2727The function sine is one of the trigonometric functions applicable on right-angled triangles. It is primarily applicable on an acute angle in a triangle wherein the sine function is mathematically represented as sinθ, for acute angle θ in a right-angled triangle. The function sinθ is calculated as the ratio of perpendicular and hypotenuse. Mathematically,
sinθ = perpendicular/hypotenuse
For right triangle ABC
sinθ = AB/AC
The original value of sin15° = (√3 – 1) / 2√2. This value of sin15° can be obtained through different methods. To solve for the sine value, the following values of sine and cosine functions for some standard angles must be considered:
The following methods can be used to derive the value of sin15°:
For finding the value of sin15° use this method. Consider the sine and cosine trigonometric functions of standard angles 60° and 45° i.e. sin45°, sin60°, cos45° and cos60°.
We know,
sin45° = 1/√2
sin60° = √3/2
cos45° = 1/√2
cos60° = ½
Proceed as below
sin15° = sin(60 – 45)…..(1)
Comparing the above equation with the formula
sin(A – B) = sinA . cosB – cosA . sinB…..(2)
Substituting the values of equation (1) in the equation (2), we get
sin(60 – 45) = (sin60 . cos45) – (cos60 . sin45)
sin15° = (√3/2 . 1/√2) – (½ . 1/√2)
sin15° = (√3 – 1) / 2√2
This can also be solved using trigonometric functions of standard angles 30° and 45°.
We know,
sin30° = ½
sin45° = 1/√2
cos30° = √3/2
cos45° = 1/√2
Substituting the above values in equation (2)
sin(45 – 30) = (sin45 . cos30) – (cos45 . sin30)
sin15° = (1/√2 . √3/2) – (1/√2 . ½)
sin15° = (√3 – 1) / 2√2
This concept is based on half angle property of trigonometric function. Considering the trigonometric functions sine and cosine for the standard angle 30°, let
(sinA/2 + cosA/2)² = (sinA/2)² + (cosA/2)² + 2 . sinA/2 . cosA/2
(sinA/2 + cosA/2)² = 1 + 2 . sinA/2 . cosA/2…..(3)
As we known,
2 sinx. cosx = sin2x…..equation (4)
Using equation (4) to reduce equation (3), we get
(sinA/2 + cosA/2)² = 1 + sinA
On taking square root both sides,
(sinA/2 + cosA/2) = ±√ (1 + sinA) ….. (5)
Now, let A = 30°, then A/2 = 15°. putting the values of A and A/2 in equation (5), we get
sin15° + cos15° = ±√ (1 + sin30°)….. (6)
Similarly, obtaining the corresponding value for (sin15° + cos15°) which comes out to be
sin15° - cos15° = ±√ (1 - sin30°)….. (7)
On adding equations (6) and (7), we get
(sin15° + cos15°) + (sin15° - cos15°) = √ (1 + sin30°) + √ (1 - sin30°)
2 . sin15° = √ (1 + sin30°) + √ (1 - sin30°)
2 . sin15° = √(1 + ½) + √(1 – ½)
2 . sin15° = √(3 -1)/ √2
sin15° = √(3 -1)/ 2√2
which is the required value.