Question

what is the value of \\(\cos\left(\frac{7\pi}{4}\
ight)\\)? \\(\circ -\frac{\sqrt{2}}{2}\\) \\(\circ -\frac{\sqrt{3}}{2}\\) \\(\circ \frac{1}{2}\\) \\(\circ \frac{\sqrt{2}}{2}\\) the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.

Explanation:

Step1: Use the cosine subtraction formula

$\cos(A - B)=\cos A\cos B+\sin A\sin B$. We know that $\frac{7\pi}{4}=2\pi-\frac{\pi}{4}$. So, $\cos(\frac{7\pi}{4})=\cos(2\pi - \frac{\pi}{4})$. According to the formula, $\cos(2\pi - \frac{\pi}{4})=\cos(2\pi)\cos(\frac{\pi}{4})+\sin(2\pi)\sin(\frac{\pi}{4})$. Since $\cos(2\pi) = 1$ and $\sin(2\pi)=0$, then $\cos(\frac{7\pi}{4})=\cos(\frac{\pi}{4})$.

Step2: Recall the value of $\cos(\frac{\pi}{4})$

We know that for an isosceles right - triangle with legs of length $1$ and hypotenuse of length $\sqrt{2}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\theta=\frac{\pi}{4}$, $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$. Also, we can use the unit - circle definition. The coordinates of the point on the unit circle corresponding to an angle of $\frac{7\pi}{4}$ are $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$, and $\cos\theta=x$ - coordinate of the point on the unit circle.

Answer:

$\frac{\sqrt{2}}{2}$