# Find the exact value of the trigonometric function at the given real number. NS

Find the exact value of the trigonometric function at the given real number.
$$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

liannemdh
We would like to find the exact value of $$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$, First, we can find the reference angle of $$\displaystyle{\frac{{{7}\pi}}{{{4}}}}$$ where $$\displaystyle{2}\pi-{\frac{{{7}\pi}}{{{4}}}}={\frac{{\pi}}{{{4}}}}$$, so the reference angle is $$\displaystyle{\frac{{\pi}}{{{4}}}}$$.
Now the next step is to define the sign of $$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$. We know that $$\displaystyle{\frac{{{7}\pi}}{{{4}}}}$$ is in quadrant 3 which the sine function is negative in this quadrant, so the value of $$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$ is negative and we can simplify it as follows:
$$\displaystyle\therefore{\sin{{\frac{{{7}\pi}}{{{4}}}}}}={\sin{{\left({2}\pi-{\frac{{\pi}}{{{4}}}}\right)}}}=-{\sin{{\frac{{\pi}}{{{4}}}}}}$$
Note that the first step was to find the reference angle and then was to define the sign of $$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$.
Now we can find the value of $$\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}$$ by knowing the value of $$\displaystyle{\sin{{\frac{{\pi}}{{{4}}}}}}$$ which equals $$\displaystyle{\frac{{\sqrt{{{2}}}}}{{{2}}}}$$
$$\displaystyle\therefore{\sin{{\frac{{{7}\pi}}{{{4}}}}}}={\sin{{\left({2}\pi-{\frac{{\pi}}{{{4}}}}\right)}}}=-{\sin{{\frac{{\pi}}{{{4}}}}}}=-{\frac{{\sqrt{{{2}}}}}{{{2}}}}$$