Question
Use the Unit Circle to find the value. Please use the Unit Circle on the Trigonometry Reference Sheet. cos7π4 (1 point) Responses 2√2 the fraction with numerator square root of 2 and denominator 2 −2√2 negative the fraction with numerator square root of 2 and denominator 2 12 1 half −12
Answers
Answered by
GPT-4o mini
To find \(\cos\left(\frac{7\pi}{4}\right)\), we can refer to the Unit Circle.
The angle \(\frac{7\pi}{4}\) is in the fourth quadrant and can also be expressed as:
\[
\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}
\]
This means it is \(45^\circ\) (or \(\frac{\pi}{4}\)) below the positive x-axis.
On the Unit Circle, the coordinates of the angle \(\frac{\pi}{4}\) are:
\[
\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
\]
Since \(\frac{7\pi}{4}\) is in the fourth quadrant, the cosine value remains positive while the sine value is negative. Thus, the coordinates for \(\frac{7\pi}{4}\) will be:
\[
\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)
\]
Therefore, the value of \(\cos\left(\frac{7\pi}{4}\right)\) is:
\[
\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
So the correct response is:
\[\frac{\sqrt{2}}{2}\]
The angle \(\frac{7\pi}{4}\) is in the fourth quadrant and can also be expressed as:
\[
\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}
\]
This means it is \(45^\circ\) (or \(\frac{\pi}{4}\)) below the positive x-axis.
On the Unit Circle, the coordinates of the angle \(\frac{\pi}{4}\) are:
\[
\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
\]
Since \(\frac{7\pi}{4}\) is in the fourth quadrant, the cosine value remains positive while the sine value is negative. Thus, the coordinates for \(\frac{7\pi}{4}\) will be:
\[
\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)
\]
Therefore, the value of \(\cos\left(\frac{7\pi}{4}\right)\) is:
\[
\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
So the correct response is:
\[\frac{\sqrt{2}}{2}\]