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...

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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lemjay's profile pic

lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

`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` .

mathieu1881's profile pic

mathieu1881 | Student, College Freshman | eNotes Newbie

Posted on

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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