Khumbo
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Cos(x +30)=sinx
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Rexicon7
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1) Expand cos(x+30) in the form cos(x+-a) = cosxsina -+ sinxcosa giving cosxsin30 - sinxcos30 = sinx
2) sin30 = 1/2 and cos30 = (sqrt3)/2 (note: sqrt = sqare root) so 1/2cosx - (sqrt3)/2sinx = sinx
3) Bring the sin values over onto one side of the eqation: 1/2 cosx = (2+sqrt3)/2 sinx.
4) (optional) Multiply both sides by 2 to remove the denominator: cosx = (2+sqrt3) sinx.
5) sinx / cosx = tanx so divide both sides by cosx: 1 = ((2+sqrt3) sinx) / cosx = (2+sqrt3) tanx.
6) Divide both sides by (2+sqrt3): 1/(2+sqrt3) = tanx.
7) Take the arctan of both sides (aka tan^-1 / inverse tan): tan^-1( 1/(2+sqrt3) ) = tan^-1(tanx) = x = 75.
8) Solve for x in the given interval. E.g. for 0 =< x <= 360, there are 2 solutions: 75 degrees and 75+180 = 255 degrees. You may find it helpful to draw the tangent graph and see where the solutions came from graphically.
Last edited by Rexicon7; 2 days ago
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Knortfoxx
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The solution given above works, but a much simpler (GCSE) method of solving this is to write out all the special angles. (e.g. cos(30), sin(45) etc. Based on the graphs of cos and sin, you should be able to identify where cos(x+30)=sin(x)
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