Solve for x. 100cos(360 - x) + 150cos(330-x) + 200cos(180 + x) = 0 100cos(x) + 150cos(30 + x ) -200cos(x) =0 I have to find the value of x but i can't get to it. I always get the wrong angle but I can't figure out what to do differently. If you draw the x value on a graph you can see that those two equations are the same exept that i changed the + for a - for the last one since it's going in the negative x. The answer is 21.7 degrees.
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`100cos(360-x)+150cos(330-x)+200cos(180+x)=0`
To simplify the equation, use the identity for sum and difference of two angles which are:
`cos(A+B)=cosAcosB-sinAsinB`
`cos(A-B)=cosAcosB+sinAsinB`
So the equation becomes,
`100(cos360cosx+sin360sinx)+150(cos330cosx+sin330sinx)+200(cos180cosx-sin180sinx)=0`
Note that c`os360=1` , `sin360=0` , `cos330=sqrt3/2` , `sin330=-1/2` , `cos180=-1` and `sin180=1` .
`100(cosx+0*sinx)+150(sqrt3/2cosx-1/2sinx)+200(-cosx+0*sinx)=0`
`100cosx+75sqrt3cosx-75sinx-200cosx=0`
Then, divide both sides by the GCF of the equation.
`(100cosx+75sqrt3cosx-75sinx-200cosx)/25=0/25`
`4cosx+3sqrt3cosx-3sinx-8cosx=0`
Combine like terms.
`3sqrt3cosx-4cosx-3sinx=0`
Then express this as one trigonometric function. Divide both sides by cosx.
`(3sqrt3cosx-4cosx-3sinx)/cosx=0/(cosx)`
`3sqrt3-4-3sinx/cosx=0`
Note that `tanx =(sinx)/(cosx)` .
`3sqrt3-4-3tanx=0`
Isolate the term with x.
`3tanx=3sqrt3-4`
`tanx=(3sqrt3-4)/3`
`x=tan^(-1)((3sqrt3-4)/3)`
`x=21.7`
Hence, the value of x in the given equation is `21.7^o` .
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Super clear answer, thank you very much. I didn't planned on using trigonometric identities but this was the way to go. You should be a teacher at the university instead of here, your clear explanations would me much appreciated. 5 stars
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