To find the value of [tex]\(\sin 45^\circ\)[/tex] or [tex]\(\sin \frac{\pi}{4}\)[/tex], we can use the known trigonometric angle values.
First, let's recall that 45 degrees or [tex]\(\frac{\pi}{4}\)[/tex] radians is a special angle in trigonometry. For this angle, both the sine and cosine have the same value. The fundamental trigonometric ratios for 45 degrees are derived from an isosceles right triangle (45-45-90 triangle), where the two legs are equal in length.
For an isosceles right triangle with legs of length [tex]\(1\)[/tex]:
- The hypotenuse can be calculated using the Pythagorean theorem:
[tex]\[
\text{hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}
\][/tex]
- The sine of 45 degrees ([tex]\(\sin 45^\circ\)[/tex]) is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
[tex]\[
\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
Therefore, we get:
[tex]\[
\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \approx 0.7071067811865475
\][/tex]
So, [tex]\(\sin 45^\circ\)[/tex] or [tex]\(\sin \frac{\pi}{4}\)[/tex] has a value of approximately [tex]\(0.7071067811865475\)[/tex].
