Question

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

Answer 1

`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`

Answer 2

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):

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`.