How do you find the exact value of sin 45 degrees?

2 Answers
Mar 28, 2018

#sin45^@=sqrt(2)/2#

Explanation:

This is a common value, in which #sin45^@=1/sqrt2#.

We can now rationalize the fraction, which comes out to:

#1/sqrt2*1/1#

#=1/sqrt2*(sqrt(2))/sqrt2#

#=sqrt(2)/2#

Mar 28, 2018

The exact value of #\sin(45)# is #\frac{\sqrt{2}}{2}#.

Explanation:

Consider #\triangle ABC# to be right-angled in #B# and choose #\angle BCA# such that its measure is #45^o#. Since the triangle is isosceles, we can deduce that angle #\angle CAB# is also #45^o#. Then pick an arbitrary value for #AB# and #BC# and apply the Pythagorean theorem. I'll go with the unit triangle, choosing both #AB# and #BC# to be #1# (remember, the triangle is isosceles):

A triangle and my great photoshop skills...

The hypothenuse #AC# can easily be calculated now: #AC=\sqrt{BC^2+AB^2}=\sqrt{1^2+1^2}=\sqrt{2}#.

The sine is defined as the ratio between the opposed side and the hypothenuse. Therefore, #\sin 45^o=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}#.

In decimal form, it is roughly #0.7071067812#.