Certainly! Let's determine the value of the trigonometric function [tex]\(\sin \left(\frac{\pi}{4}\right)\)[/tex].
First, recall the unit circle and fundamental trigonometric identities. The angle [tex]\(\frac{\pi}{4}\)[/tex] radians corresponds to 45 degrees. For this angle, we can use the fact that in a right triangle where both non-hypotenuse sides are equal (an isosceles right triangle), each of the non-right angles is 45 degrees.
For an isosceles right triangle with legs of length 1, the hypotenuse can be found using the Pythagorean theorem:
[tex]\[
\text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}
\][/tex]
In such a triangle, the sine of a 45-degree angle (or [tex]\(\frac{\pi}{4}\)[/tex] radians) is the length of one leg divided by the hypotenuse:
[tex]\[
\sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
\][/tex]
Rationalizing the denominator, we multiply the numerator and denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\sin \left(\frac{\pi}{4}\right) = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
Hence, the exact value of [tex]\(\sin \left(\frac{\pi}{4}\right)\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
When we calculate the numerical value of this expression, we get approximately:
[tex]\[
\sin \left(\frac{\pi}{4}\right) \approx 0.7071067811865475
\][/tex]
Therefore, the exact value of the trigonometric function [tex]\(\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex] and numerically it approximates to 0.7071067811865475.
