use identities to find values of the sine and cosine functions for the angle measure. \nθ, given that ( cos…

use identities to find values of the sine and cosine functions for the angle measure. \nθ, given that ( cos 2 \theta=\frac{56}{65} ) and ( 90^{circ}<\theta<180^{circ} )\n( sin \theta=\frac{3 sqrt{130}}{130} )\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\n( cos \theta=square )\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

use identities to find values of the sine and cosine functions for the angle measure. \nθ, given that ( cos 2 \theta=\frac{56}{65} ) and ( 90^{circ}<\theta<180^{circ} )\n( sin \theta=\frac{3 sqrt{130}}{130} )\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\n( cos \theta=square )\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Answer

Explanation:

Step1: Use the double - angle identity for cosine

The double - angle identity for cosine is (\cos2\theta = 1 - 2\sin^{2}\theta=2\cos^{2}\theta - 1). We can also use the Pythagorean identity (\sin^{2}\theta+\cos^{2}\theta = 1). Since we know (\cos2\theta=\frac{56}{65}), and we want to find (\cos\theta), we use the identity (\cos2\theta=2\cos^{2}\theta - 1). Rearrange the identity to solve for (\cos^{2}\theta): (\cos^{2}\theta=\frac{1 + \cos2\theta}{2}) Substitute (\cos2\theta=\frac{56}{65}) into the formula: (\cos^{2}\theta=\frac{1+\frac{56}{65}}{2}=\frac{\frac{65 + 56}{65}}{2}=\frac{121}{130})

Step2: Determine the sign of (\cos\theta)

Given that (90^{\circ}<\theta<180^{\circ}), (\theta) is in the second quadrant. In the second quadrant, the cosine function is negative. Take the square root of (\cos^{2}\theta=\frac{121}{130}). So (\cos\theta=-\sqrt{\frac{121}{130}}) Rationalize the denominator: (\cos\theta=-\frac{11}{\sqrt{130}}=-\frac{11\sqrt{130}}{130})

Answer:

(\cos\theta =-\frac{11\sqrt{130}}{130})