Determine the sign of sin(5π4) and the quadrant in which it lies.(1 point)

Question

Responses

sin(5π4) is negative and lies in Quadrant IV.
sine left parenthesis Start Fraction 5 pi over 4 End Fraction right parenthesis is negative and lies in Quadrant IV.

sin(5π4) is positive and lies in Quadrant III.
sine left parenthesis Start Fraction 5 pi over 4 End Fraction right parenthesis is positive and lies in Quadrant III.

sin(5π4) is positive and lies in Quadrant II.
sine left parenthesis Start Fraction 5 pi over 4 End Fraction right parenthesis is positive and lies in Quadrant II.

sin(5π4) is negative and lies in Quadrant III.

Answer 1

To determine the sign of \(\sin\left(\frac{5\pi}{4}\right)\) and the quadrant in which it lies, let's analyze the angle \(\frac{5\pi}{4}\).

1. Convert to degrees: \[ \frac{5\pi}{4} \times \frac{180^\circ}{\pi} = \frac{5 \times 180^\circ}{4} = 225^\circ \] This angle is in standard position.

2. Identify the quadrant: An angle of \(225^\circ\) lies in Quadrant III, where both sine and cosine are negative.

3. Determine the sign of \(\sin\) in this quadrant: Since \(\frac{5\pi}{4}\) is located in Quadrant III, the sine function is negative in this quadrant.

Based on this analysis, the correct response is:

sin(5π4) is negative and lies in Quadrant III.