Questions: Select the correct choice below and fill in any answer boxes within your choice. A. sin(-5π/4)=□ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. cos(-5π/4)= □ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined.

Solution Steps

• The sine function is odd, meaning \(\sin(-x) = -\sin(x)\).
• Therefore, \(\sin\left(-\frac{5\pi}{4}\right) = -\sin\left(\frac{5\pi}{4}\right)\).
• The angle \(\frac{5\pi}{4}\) lies in the third quadrant, where sine is negative.
• The reference angle for \(\frac{5\pi}{4}\) is \(\frac{\pi}{4}\), and \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
• Thus, \(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), and \(\sin\left(-\frac{5\pi}{4}\right) = -\left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}\).

• The cosine function is even, meaning \(\cos(-x) = \cos(x)\).
• Therefore, \(\cos\left(-\frac{5\pi}{4}\right) = \cos\left(\frac{5\pi}{4}\right)\).
• The angle \(\frac{5\pi}{4}\) lies in the third quadrant, where cosine is negative.
• The reference angle for \(\frac{5\pi}{4}\) is \(\frac{\pi}{4}\), and \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
• Thus, \(\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), and \(\cos\left(-\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

• For \(\sin\left(-\frac{5\pi}{4}\right)\), the correct choice is A, and the value is \(\frac{\sqrt{2}}{2}\).
• For \(\cos\left(-\frac{5\pi}{4}\right)\), the correct choice is A, and the value is \(-\frac{\sqrt{2}}{2}\).

Final Answer