How do i find the exact value of sin 15 using the half angle formula
sin (30/2) = +/- sqrt [(1-cos 30)/2]
+ in first quadrant
= sqrt [ (1 - {sqrt 3}/2)/2) ]
= sqrt [ (2 - sqrt 3)/4 ]
= (1/2) sqrt (2 - sqrt 3)
use
cos 2A = 1 - 2sin^2 A
cos 30° = 1 - 2sin^2 15°
√3/2 = 1- 2sin^2 15°
2sin^2 15° - 1 - √3/2 = (2-√3)/2
sin^2 15° = (2-√3)/4
sin 15° = √(2-√3) /2
it would have been easier to do
sin15°
= sin(45°-30°)
= sin45cos30 - cos45sin30
= (√2/2)(√3/2) -(√2/2)(1/2)
= (√6 - √2)/4 which yields the same result as above
To find the exact value of sin 15 using the half angle formula, you can follow these steps:
Step 1: Recognize that sin 15 is not directly given in any special right triangle.
Step 2: Since sin 30 is a known value (equal to 1/2), you can use the half angle formula for sine, which states that sin (θ/2) = √[(1 - cos θ) / 2].
Step 3: In order to use the half angle formula, find the value of cos 30. By using a special right triangle (a 30-60-90 triangle), you can determine that cos 30 is equal to √3/2.
Step 4: Plug the value of cos 30 into the half angle formula for sin (θ/2):
sin (15/2) = √[(1 - cos 30) / 2]
= √[(1 - (√3/2)) / 2]
= √[(2 - √3) / 4]
Step 5: Simplify the expression, rationalizing the denominator if necessary:
sin (15/2) = (√(2 - √3)) / 2
So, the exact value of sin 15 using the half angle formula is (√(2 - √3)) / 2.