Question

Apply the sum and difference formulas for sine to simplify the left-hand side of the equation.

http://gyazo.com/d2303bff1c440dc0c6fc7caf6d79e433

Answer 1

At first, I would convert radians to degrees, since I like degrees better.

\(\normalsize\color{blue}{\sin(x+45)-\sin(x-45)=1\LARGE\color{white}{ \rm │ }}\)

Answer 2

then applying sin(a+b) and sin(a-b) rule, but recall that sin(45)=cos(45)...(since they both are equivalent to `√2 /2` )

\(\normalsize\color{blue}{\sin(x+45)-\sin(x-45)=1\LARGE\color{white}{ \rm │ }}\)
\(\normalsize\color{blue}{\sin(x)\cos(45)+\sin(45)\cos(x)-[~~\sin(x)\cos(45)-\sin(45)\cos(x)~~]=1\LARGE\color{white}{ \rm │ }}\)
\(\normalsize\color{blue}{\sin(x)\cos(45)+\sin(45)\cos(x)-\sin(x)\cos(45)+\sin(45)\cos(x)=1\LARGE\color{white}{ \rm │ }}\)
\(\normalsize\color{blue}{\sin(45)\cos(x)+\sin(45)\cos(x)=1\LARGE\color{white}{ \rm │ }}\)
\(\normalsize\color{blue}{2\sin(45)\cos(x)=1\LARGE\color{white}{ \rm │ }}\) this is what I got so far.

Answer 3

\(\normalsize\color{blue}{2\sin(45)\cos(x)=1\LARGE\color{white}{ \rm │ }}\)

\(\normalsize\color{blue}{2(\frac{\sqrt{2}}{2})\cos(x)=1\LARGE\color{white}{ \rm │ }}\)

\(\normalsize\color{blue}{\sqrt{2}~\cos(x)=1\LARGE\color{white}{ \rm │ }}\)

Answer 4

Do you need to solve for x?

Divide both sides by √2,
Multiply top and bottom of the fraction on the right side, times √2 to rationalize the denominator, and you get cos(x)=√2 /2 which means that cos=45º (first solution)

Answer 5

Yes, I also need to solve for x :o