(a) Using the Law of Sines, solve for the value $\sin(A)$ given that $x=7\sqrt{2}$, $y=7\sqrt{3}$, and $C=45^\circ$.

(b) Using the Law of Sines, solve for the value $\sin(B)$ given that $x=7\sqrt{2}$, $y=7\sqrt{3}$, and $C=45^\circ$.

(c) How are the values of $\sin(A)$ and $\sin(B)$ related?

A. $\sin(A)=\frac{2}{3}\sin(B)$

B. $\frac{\sin(A)}{\sqrt{3}}=\frac{\sin(B)}{\sqrt{2}}$

C. $\sin(A)$ is smaller than $\sin(B)$

D. $\sin(A)=\sin(B)$.

E. $\sin(A)=\frac{3}{2}\sin(B)$

F. None of the above

How is this possible?

A. The law of sines allows us to conclude $\frac{\sin(A)}{\sqrt{3}}=\frac{\sin(B)}{\sqrt{2}}$

B. Because $A$ is smaller than $B$, then $\sin(A)<\sin(B)$.

C. For $0<y<1$, there is always exactly one value of $\theta$ between $0$ and $\pi$ such that $\sin(\theta)=y$.

D. For $0<y<1$, there are always two values of $\theta$ between $0$ and $\pi$ such that $\sin(\theta)=y$.

E. None of the above

(d) Using your knowledge of inverse trig functions and the illustration of each triangle, solve for $A$ and $B$.

(e) Based on your work above, what is a weakness of the Law of Sines?

A. When using it to solve for an angle $\theta$, you will reduce to an equation $\sin(\theta)=y$ for some $0<y<1$. There are two possible solutions to this, but your calculator will only give you one.

B. When using it to solve for an angle $\theta$, you will often reduce to an equation $\sin(\theta)=y$ for some $y>1$, and there are no possible solutions to this.

C. We need to use the picture of the triangle to decide what our final answer should be.

D. When using it to solve for an angle $\theta$, you will often reduce to an equation $\sin(\theta)=\sin(\phi)$ for some angle $\phi$, and this implies $\theta=\phi$.

E. When using it to solve for an angle $\theta$, you will often reduce to an equation $\cos(\theta)=y$ for some $y>1$, and there are no possible solutions to this.

F. None of the above