How do you find the exact values of sin 15 degrees using the half angle formula?

3 Answers
Jul 21, 2015

I found: sin(15°)=0.258

Explanation:

Using the Half Angle Formula:
color(red)(sin^2(x)=1/2[1-cos(2x)])

with x=15° and 2x=30°

you get:

sin^2(15°)=1/2[1-cos(30°)]

knowing that: cos(30°)=sqrt(3)/2:

sin^2(15°)=1/2[1-sqrt(3)/2]
sin^2(15°)=(2-sqrt(3))/4=0.067

So:
sin(15°)=+-sqrt(0.067)=+-0.258
We choose the positive one.

May 14, 2017

sin(15^@)=sqrt(2-sqrt(3))/2=(sqrt(6)-sqrt(2))/4
See below.

Explanation:

Find exact value of sin(15^@) with half-angle formula.

Consider the half-angle formula for sine: sin(theta/2)=sqrt((1-cosx)/2)

Since we know that 15 is half of 30, we can plug 30^@ in as theta and simplify:
sin(15^@)=sin(30^@/2)=sqrt((1-cos(30^@))/2)
=sqrt((1-sqrt(3)/2)/2)
=sqrt(((2-sqrt(3))/2)/2)
=sqrt((2-sqrt(3))/4)
or see slightly more advanced method to remove nested root (at the bottom)
=sqrt(2-sqrt(3))/2
which is our answer, but has a nested root
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Removing nested root from answer:
Let us multiply both the numerator and denominator inside the square root by 2:
=sqrt((2-sqrt(3))/4*2/2)
=sqrt((4-2sqrt(3))/8)

Now we can write 4-2sqrt(3) as a square in the numerator:
=sqrt((sqrt(3)-1)^2/8)

We can take out a sqrt(3)-1 from the numerator and a 2 from the denominator:
=((sqrt(3)-1)/2)*1/sqrt(2)
=(sqrt(3)-1)/(2sqrt(2))

We can rationalize the denominator:
=(sqrt(2)(sqrt(3)-1))/4
=(sqrt(6)-sqrt(2))/4

which is our answer without a nested root.

Aug 5, 2018

(sqrt6-sqrt2)/4

Explanation:

Use

sin(A-B) = sinA*cosB - SinB*cosA

So

sin 15 = sin(45-30) = sin45*cos30 - sin30*cos45

We know

sin45 = cos45 = (sqrt2)/2

We also know

cos30 = (sqrt3)/2

sin30 = 1/2

Plugging in the values, we get

sin15 = (sqrt2)/2*((sqrt3)/2-1/2)

Simplifying, we get

(sqrt6-sqrt2)/4