# precalculus

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Solve 7 cos(2 theta) = 7 sin^2(theta) + 3 for all solutions 0 less than or equal to theta less than 2pi.

Solve 7 sin(2 w) - 4 cos(w) =0 for all solutions 0 le w lt 2pi

• precalculus -

le= less than or equal to
lt = less than

I keep getting numbers that are not on the unit circle and then when I try to plug them into my calculator I'm getting really big numbers. Please help. I've tried looking in my book but there are no problems like these.

• precalculus -

7 cos(2 theta) = 7 sin^2(theta) + 3
7 cos^2 Ø - 7sin^2 Ø = 3
cos^2Ø - sin^2 Ø = 3/7
cos 2Ø = 3/7

so 2Ø is in I or in IV
2Ø = 1.1279 or 2Ø = 2π-1.1278 = 5.1553
Ø = .5639 or Ø = 2.578
Now the period of cos 2Ø = π , so by adding multiples of π to any answer will yield a new answer

so Ø = .5639+π = 3.7055
Ø = 5.1553+π = 5.71924

7 sin 2w - 4cos w = 0
7(2sinwcosw) - 4cosw = 0
cosw(14sinw - 4 ) = 0
cosw = 0 or sinw = 2/7

if cosw = 0, then w = π/2 , 3π/2
if cosw = 2/7 , then w = 1.281 or 2π - 1.281 = 5.002

• precalculus -

Thank you for your help! My numbers were way off haha.

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