Hi! Can someone help check this for me and see if I'm doing it right? Thanks!! :)
Directions: Use the Half-Angle formulas to determine the exact value of sin(pi/12). Here's what I have:
π/12 = ( 180° ) / 12 = 15°.
= sin ( π/12 )
= sin 15°
= sin ( 45° - 30°)
= sin 45°· cos 30° - cos 45°· sin 30°
= (1/√2)·(√3 /2 ) - (1/√2)·(1/2)
= ( √3 - 1 ) / (2√2)
Even though you got the correct answer you did not follow the required method.
(That's like winning a 100 m backstroke race by using the butterfly stroke , you would be DQ'd)
It said to use the Half-Angle formula
There are two,
sin (2A) = 2sinAcosA
and
cos(2A) = cos^2 A - sin^2 A = 1 - 2sin^2 A = 2cos^2 A - 1
I will use
cos(2A) = 1 - 2sin^2 A
cos 30° = 1 - 2sin^2 15°
2 sin^2 15 = 1 - √3/2 = (2-√3)/2
sin^2 15 = (2 - √3)/4
sin 15 = √(2 - √3) /2
Check: by using your calculator to find sin15 , and evealutationg my answer to show they are equal
even though this answer looks different from yours, use your calculator to show that it is the same as your answer
I interpret the instruction to use the Half-Angle formula as
sin(15°) = √((1-cos30°)/2)
= √((1-√3/2)/2
= √(2-√3)/2
= (√6-√2)/2
Granted, the half- and double-angle formulas are two sides of the same coin, but ...
To verify if your solution is correct, we can use the Half-Angle formula for sine. The Half-Angle formula states that:
sin(x/2) = ±√((1 - cos(x)) / 2)
In this case, x is π/6 (30°), since we want to find sin(π/12).
To use the Half-Angle formula, we find cos(π/6) (30°) first:
cos(π/6) = √3/2
Now, we substitute this value into the formula:
sin(π/12) = ±√((1 - √3/2) / 2)
Next, we simplify the expression:
sin(π/12) = ±√((2 - √3) / 4)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator:
sin(π/12) = ±√((2 - √3) / 4) * (√4 / √4)
This simplifies to:
sin(π/12) = ±√(2 - √3) / 2
Comparing this result with your answer, ( √3 - 1 ) / (2√2), we can see that your solution is incorrect. The correct result using the Half-Angle formula for sin(π/12) is ±√(2 - √3) / 2.