A-Level Maths: E7-37 [Trig Equations: Solve 5tan^2(x) - 38tan(x) - 16 = 0, 0-360 degrees]

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Hi. Why do you sometimes add 45 as the period and other times add 180. Same with cos and sin, why is sometimes 360 and other times its 120. How do you know which one to do?

a.a
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how come there is a solution for tanx = 8. would there not be any solutions because it is above 1. just like how tanx= -3 would have no solutions because it is smaller than -1?

hannahk