Can you please solve and determine all solutions of this equation in radian measure?

Question

cos (theta) + 6 sec (theta) = - 7

Answer 1

cos - 6 sec = -7
cos - 6/cos = -7
(cos^2 -6)/cos = -7 cos/cos
cos^2 - 6 = -7 cos
cos^2 + 7 cos -6 = 0

cos = [ -7 +/- SQRT( 49 - 24) ] / 2
cos = [ -7 +/- 5 ] / 2

cos = -1 or -6

In real numbers, cos(theta) = -6 is impossible.

So we only have cos(theta) = -1 to solve.

Theta = arc-cos(-1) = half a circumference

(where circumference = 2*pi*r)

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Because cos is cyclical, it returns to the same value after every circumference (2*pi).

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Theta = pi + 2*pi*k
where k is any integer.