# The question asks for the exact value of the trignometric function at the given

The question asks for the exact value of the trignometric function at the given real number:
$$\displaystyle{\sin{{\left({\frac{{{7}\pi}}{{{4}}}}\right)}}}$$ I answered $$\displaystyle{\left({\frac{{-\sqrt{{{2}}}}}{{{2}}}}\right)}$$

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avortarF
To find the exact value of $$\displaystyle{\sin{{\left({\frac{{{7}\pi}}{{{4}}}}\right)}}}$$
Using trigonometry table and trigonometry unit circle
$$\displaystyle{\sin{{\left({\frac{{{7}\pi}}{{{4}}}}\right)}}}={\sin{{\left(-{\frac{{\pi}}{{{4}}}}+{\frac{{{8}\pi}}{{{4}}}}\right)}}}$$
$$\displaystyle={\sin{{\left(-{\frac{{\pi}}{{{4}}}}+{2}\pi\right)}}}$$
$$\displaystyle={\sin{{\left(-{\frac{{\pi}}{{{4}}}}\right)}}}$$
$$\displaystyle=-{\sin{{\left({\frac{{\pi}}{{{4}}}}\right)}}}$$
$$\displaystyle=-{\frac{{\sqrt{{{2}}}}}{{{2}}}}$$