Question

The question asks for the exact value of the trigonometric function at the given real number:
\(\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{4}}\right)}}}\)

Answer 1

Trigonometric formula:
\(\displaystyle{\sin{{\left(\pi-\theta\right)}}}={\sin{\theta}}\)
Solving the given trigonometric function.
\(\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{4}}\right)}}}\)
\(\displaystyle={\sin{{\left(\pi-\frac{\pi}{{4}}\right)}}}\)
\(\displaystyle={\sin{{\left(\frac{\pi}{{4}}\right)}}}{\left\lbrace,'{\sin{{\left(\pi-\theta\right)}}}={\sin{\theta}}\right\rbrace}\)
\(\displaystyle=\frac{{1}}{\sqrt{{2}}}\)
The value of given trigonometric function is shown below.
\(\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{4}}\right)}}}=\frac{{1}}{\sqrt{{2}}}\)