use the half - angle formula to find the exact value of the expression sin(-5π/8). (simplify your answer…

use the half - angle formula to find the exact value of the expression sin(-5π/8). (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) a. sin(-5π/8)=√((1 - sin())/2) b. sin(-5π/8)=-√((1 - cos(-5π/4))/2) c. sin(-5π/8)=√((1 - cos())/2) d. sin(-5π/8)=√((1 + cos())/2) e. sin(-5π/8)=-√((1 + cos())/2) f. sin(-5π/8)=-√((1 + sin())/2) g. sin(-5π/8)=-√((1 - sin())/2) h. sin(-5π/8)=√((1 + sin())/2) the exact value of the trigonometric function sin(-5π/8) is -√((1 + √2/2)/2). (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

use the half - angle formula to find the exact value of the expression sin(-5π/8). (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) a. sin(-5π/8)=√((1 - sin())/2) b. sin(-5π/8)=-√((1 - cos(-5π/4))/2) c. sin(-5π/8)=√((1 - cos())/2) d. sin(-5π/8)=√((1 + cos())/2) e. sin(-5π/8)=-√((1 + cos())/2) f. sin(-5π/8)=-√((1 + sin())/2) g. sin(-5π/8)=-√((1 - sin())/2) h. sin(-5π/8)=√((1 + sin())/2) the exact value of the trigonometric function sin(-5π/8) is -√((1 + √2/2)/2). (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Recall half - angle formula for sine

The half - angle formula for $\sin\frac{\alpha}{2}$ is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$. Let $\frac{\alpha}{2}=-\frac{5\pi}{8}$, then $\alpha=-\frac{5\pi}{4}$.

Step2: Determine the sign

The angle $-\frac{5\pi}{8}$ is in the third - quadrant where sine is negative. So $\sin(-\frac{5\pi}{8})=-\sqrt{\frac{1-\cos(-\frac{5\pi}{4})}{2}}$. We know that $\cos(-\frac{5\pi}{4})=\cos\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}$.

Step3: Substitute the value of cosine

Substitute $\cos(-\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$ into the formula: [ \begin{align*} \sin(-\frac{5\pi}{8})&=-\sqrt{\frac{1-(-\frac{\sqrt{2}}{2})}{2}}\ &=-\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}\ &=-\sqrt{\frac{2+\sqrt{2}}{4}}\ &=-\frac{\sqrt{2 + \sqrt{2}}}{2} \end{align*} ] The correct option for the formula application is B. $\sin(-\frac{5\pi}{8})=-\sqrt{\frac{1-\cos(-\frac{5\pi}{4})}{2}}$.

Answer:

B. $\sin(-\frac{5\pi}{8})=-\sqrt{\frac{1-\cos(-\frac{5\pi}{4})}{2}}$, and the exact value of $\sin(-\frac{5\pi}{8})$ is $-\frac{\sqrt{2+\sqrt{2}}}{2}$