Question

need help with this ASAP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 1

`(120)/(169)`

Explanation:

Find the exact value of the given trigonometric expression.

`\sin(2\sin^(-1)((12)/(13) ))= \ ??\\\\\hrule`

The method I am about to show you will allow you to complete this problem without the use of a calculator. Although, memorization of the trigonometric identities is essential.

Apply the following double-angle identity.

`\boxed{\left\begin{array}{ccc}\text{\underline{Double-Angle Identity:}}\\\\\sin(2A)=2\sin(A)\cos(A)\end{array}\right}`

`\sin(2\sin^(-1)((12)/(13) ))\\\\\\ \\\Longrightarrow 2\sin(\sin^(-1)((12)/(13) ))\cos(\sin^(-1)((12)/(13) ))\\\\\\ \\\Longrightarrow 2\Big((12)/(13)\Big)\Big(\sqrt{1-((12)/(13) )^2} \Big)\\\\\\ \\\Longrightarrow 2\Big((12)/(13)\Big)\Big(\sqrt{1-((12)/(13) )^2} \Big)\\\\\\ \\\Longrightarrow 2\Big((12)/(13)\Big)\Big(\sqrt{1-(144)/(169)}} \Big)\\\\\\ \\\Longrightarrow 2\Big((12)/(13)\Big)\Big(\sqrt{(25)/(169)}} \Big)`

`\Longrightarrow 2\Big((12)/(13)\Big)\Big((5)/(13) } \Big)\\\\\\\\\Longrightarrow \boxed{\boxed{(120)/(169) }}`

Thus, the problem is solved.