Use the information given about the angles a and b to find the exact value of: $\cos (\alpha+\beta)$. $\sin \alpha=-\frac{3}{5}, \pi<\alpha<\frac{3 \pi}{2} ; \cos \beta=\frac{12}{13}, \frac{3 \pi}{2}<\beta<2 \pi$

First use (20) to express the general solution of the given differential equation in terms of Bessel functions. Then use (26) and (27) to express the general solution in terms of elementary functions. $16 x^{2} y^{\prime \prime}+32 x y^{\prime}+\left(x^{4}-12\right) y=0$

Determine the exact value for each trigonometric expression given $\sin \alpha=\frac{5}{13}, \sin \beta=-\frac{3}{5}, \frac{\pi}{2} \leq \alpha \leq \pi$, and $\frac{3 \pi}{2} \leq \beta \leq 2 \pi$.

$$\cos (\alpha+\beta)$$

Use the information given about the angles $\alpha$ and $\beta$ to find the exact value of: $(a)\ \sin (\alpha+\beta)$ $(b)\ \cos (\alpha+\beta)$ $(c)\ \sin (\alpha-\beta)$ $(d)\ \tan (\alpha+\beta)$ $(e)\ \sin (2 \alpha)$ $(f)\ \cos (2 \beta)$ $(g)\ \sin \frac{\beta}{2}$ $(h)\ \cos \frac{\alpha}{2}$ $\tan \alpha=-\frac{4}{3}, \frac{\pi}{2}<\alpha<\pi; \cot \beta=\frac{12}{5}, \pi<\beta<\frac{3 \pi}{2}$

Find the exact value of each of the following under the given conditions: $(a) \sin (\alpha+\beta) \quad (b) \cos (\alpha+\beta) \quad (c) \sin (\alpha-\beta) \quad (d) \tan (\alpha-\beta)$ $\cos \alpha=\frac{\sqrt{5}}{5}, 0<\alpha<\frac{\pi}{2}; \sin \beta=-\frac{4}{5},-\frac{\pi}{2}<\beta<0$

Use the following conditions to solve

$$\sin \alpha = \frac { 4 } { 5 } , \frac { \pi } { 2 } < \alpha < \pi$$

$$\cos \beta = \frac { 5 } { 13 } , 0 < \beta < \frac { \pi } { 2 }$$

Find the exact value of each of the following.

$$\cos ( \alpha + \beta )$$

Find the exact value of each of the following under the given conditions: $(a) \sin (\alpha+\beta) \quad (b) \cos (\alpha+\beta) \quad (c) \sin (\alpha-\beta) \quad (d) \tan (\alpha-\beta)$ $\sin \alpha=\frac{3}{5}, 0<\alpha<\frac{\pi}{2}; \cos \beta=\frac{2 \sqrt{5}}{5},-\frac{\pi}{2}<\beta<0$

Find the exact value of each of the following under the given conditions: (a) $\sin (\alpha+\beta)$ (b) $\cos (\alpha+\beta)$ (c) $\sin (\alpha-\beta)$ (d) $\tan (\alpha-\beta)$, $\cos \alpha=\frac{\sqrt{5}}{5}, 0<\alpha<\frac{\pi}{2}$, $\sin \beta=-\frac{4}{5},-\frac{\pi}{2}<\beta<0$

Find the exact value for the given $\cos \alpha=\frac{4}{5}, \sin \beta=-\frac{5}{13}, 0 \leq \alpha \leq \frac{\pi}{2}$, and $\pi\le\beta\le\frac{3 \pi}{2}$,

$$\sin (\alpha+\beta)$$

Find the exact value for the given $\cos \alpha=\frac{4}{5}, \sin \beta=-\frac{5}{13}, 0 \leq \alpha \leq \frac{\pi}{2}$, and $\pi\le\beta\le\frac{3 \pi}{2}$,

$$\tan (\alpha+\beta)$$

Use the following conditions to solve

$$\sin \alpha = \frac { 4 } { 5 } , \frac { \pi } { 2 } < \alpha < \pi \\ \cos \beta = \frac { 5 } { 13 } , 0 < \beta < \frac { \pi } { 2 }$$

Find the exact value of each of the following.

$$\sin 2 \alpha$$

Use the information given about the angles a and b to find the exact value of: $\cos \frac{\alpha}{2}$. $\sin \alpha=\frac{4}{5}, 0<\alpha<\frac{\pi}{2} ; \sin \beta=\frac{5}{13}, \frac{\pi}{2}<\beta<\pi$

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Concerning the Fibonacci sequence, prove that $\sum_{k=1}^{n} F_{k}^{2}=F_{n} F_{n+1}$.

Suppose $\sin \alpha=\frac{3}{5}$ and $\cos \beta=\frac{5}{13}$. where $0<\alpha<\beta<\frac{\pi}{2}$. Find:

a. $\sin (\alpha+\beta)$

b. $\cos (\alpha+\beta)$

c. $\tan (\alpha+\beta)$