Why are they using Cosine instead of Sine for Cross Product?

[yashboi123]

Homework Statement

What is the cross product of A X C?

Relevant Equations

A X B = ABSin(x)

I don't understand why they are using cos and putting a negative in front of the answer, and secondly why they are using the 25 degree angle. The way I was thinking of solving it would be (96.0 m^2)sin(295). Can anyone explain this for me?

 

[andrewkirk]

The answer is -96 times sine of angle AOC.

The minus sign arises from the right-hand rule. Take your right hand, point the thumb in the direction of the positive z axis, and curl your fingers. The direction your fingers curl, which is anti-clockwise as we look at the diagram, must be the direction from the vector that's the first argument to the cross product to the vector that's the second argument, in order to get a positive sign. Since in this diagram $\vec A$ is clockwise from $\vec C$ we get a negative sign.

Now what about that cos? The diagram does not mark angle AOC, but we know it it is 90 degrees minus the marked 25 degree angle.
So they just use the formula for sine of the sum of two angles, as follows:
\begin{align*}
\sin\ AOC &= \sin(90 - 25) \\&= \sin(90 + (-25))\\& = \sin 90\ \cos (-25) + \cos 90\ \sin(-25)
\\&= \cos(-25)\times 1 + 0\times \sin(-25)
\\&= \cos\ 25+0
\\&= \cos\ 25\end{align*}

 

[haruspex]

@andrewkirk , I would think @yashboi123 is looking for what is wrong with sin(295), following a standard method, rather than for an alternate method. Which method the book used is unknown.

I don't understand why they are using cos and putting a negative in front of the answer, and secondly why they are using the 25 degree angle. The way I was thinking of solving it would be (96.0 m^2)sin(295). Can anyone explain this for me?

As your calculator will tell you, sin(295)=-cos(25).
There are some useful formulas:
cos(90-x)=sin(x)=sin(180-x)=-sin(-x)=sin(360+x)
So sin(295)=sin(360-65)=sin(-65)=-sin(65)=-cos(25).