Questions: Without using a calculator, compute the sine and cosine of 7π/4 by using the reference angle. What is the reference angle? radians. In what quadrant is this angle? (answer 1, 2, 3, or 4) sin(7π/4)= cos(7π/4)=

Solution Steps

To solve this problem, we need to determine the reference angle for \(\frac{7\pi}{4}\) and identify the quadrant in which this angle lies. Then, we can use the reference angle to find the sine and cosine values.

1. Reference Angle: The reference angle is the acute angle formed with the x-axis. For angles in radians, we can subtract \(2\pi\) until the angle is within the range \([0, 2\pi)\). For \(\frac{7\pi}{4}\), we subtract \(2\pi\) to find the equivalent angle within one full rotation.
2. Quadrant: Determine the quadrant by checking the range in which the angle lies.
3. Sine and Cosine Values: Use the reference angle to find the sine and cosine values, considering the signs based on the quadrant.

The given angle is \( \frac{7\pi}{4} \). To find the reference angle, we note that \( \frac{7\pi}{4} \) is already within the range \( [0, 2\pi) \).

The reference angle \( \theta_{\text{ref}} \) for \( \frac{7\pi}{4} \) can be calculated as: \[ \theta_{\text{ref}} = 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \]

The angle \( \frac{7\pi}{4} \) lies in the fourth quadrant, where angles are between \( \frac{3\pi}{2} \) and \( 2\pi \).

Using the reference angle \( \frac{\pi}{4} \): \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] Since \( \frac{7\pi}{4} \) is in the fourth quadrant, the sine value will be negative: \[ \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Final Answer