1. Hmm. I may have started you down a path that's more work. Try eliminating the trig functions. How, might you ask? By solving each equation for the trig function, squaring everything in sight, adding one to the other to get 1. So, for example,

$\cos(\theta)=\frac{1}{800}\left(S\cos(80^{\circ})-70\cos(100^{\circ})\right)$
$\sin(\theta)=\dots$

Then, set $\cos^{2}(\theta)+\sin^{2}(\theta)=1$,

and solve for S. I think that might be easier.

2. What is the result of :

S cos^2(80) + S sin^2(80) = ?

when you square the whole thing do you do it like this:

S^2cos^2(80) - 70^2cos^2(100) ?
800

3. When you square a sum, you have to foil it out: (a+b)(a+b)=a^2+2ab+b^2.

4. We forgot to convert the bearings to angles, it should turn out like this:
cos(θ) = (S cos(10) - 70 cos(10)) 1 / 800
sin(θ) = (S sin(10) + 70 sin(10)) 1 / 800

100 - 90 = 10 (the y value of vector V is negative)
90 - 80 = 10

5. In any navy, they use north = 0 degrees, and the angles increase in a clockwise fashion. In mathematics, east = 0 degrees, and the angles increase in a counter-clockwise fashion. Which are you using?

6. I'm using mathematics.

Btw, the answer I got is S = 541.45km/h

7. Is that right?

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