Question
Evaluate the function without using a calculator cos 315^•
Answers
GPT-4o mini
To evaluate \(\cos 315^\circ\), we can use properties of the unit circle and trigonometric angles.
1. **Reference Angle**: The angle \(315^\circ\) is in the fourth quadrant. The reference angle can be found by subtracting \(315^\circ\) from \(360^\circ\):
\[
360^\circ - 315^\circ = 45^\circ
\]
2. **Cosine in the Fourth Quadrant**: In the fourth quadrant, the cosine function is positive.
3. **Value of Cosine at the Reference Angle**: The cosine of the reference angle \(45^\circ\) is:
\[
\cos 45^\circ = \frac{\sqrt{2}}{2}
\]
4. **Conclusion for \(\cos 315^\circ\)**: Since \(\cos 315^\circ\) uses the reference angle \(45^\circ\) and is positive in the fourth quadrant, we have:
\[
\cos 315^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}
\]
Thus, the value of \(\cos 315^\circ\) is \(\frac{\sqrt{2}}{2}\).
1. **Reference Angle**: The angle \(315^\circ\) is in the fourth quadrant. The reference angle can be found by subtracting \(315^\circ\) from \(360^\circ\):
\[
360^\circ - 315^\circ = 45^\circ
\]
2. **Cosine in the Fourth Quadrant**: In the fourth quadrant, the cosine function is positive.
3. **Value of Cosine at the Reference Angle**: The cosine of the reference angle \(45^\circ\) is:
\[
\cos 45^\circ = \frac{\sqrt{2}}{2}
\]
4. **Conclusion for \(\cos 315^\circ\)**: Since \(\cos 315^\circ\) uses the reference angle \(45^\circ\) and is positive in the fourth quadrant, we have:
\[
\cos 315^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}
\]
Thus, the value of \(\cos 315^\circ\) is \(\frac{\sqrt{2}}{2}\).