Find the sum of all values of \(\theta\) such that \(\theta\in[0,2\pi]\) and \(\cos(\theta)+4\sec(\theta)=8 \)
+128448
i'm using x in place of theta
cos (x) + 4 sec(x) = 8
cos (x) + 4 / cos(x) = 8 multiply through by cos (x)
cos^2x + 4 = 8cos(x)
cos^2 (x) - 8cos(x) + 4 = 0
Let cos(x) = m and we have that
m^2 - 8m + 4 = 0
m^2 - 8m = - 4 complete the square on m
m^2 - 8m + 16 = -4 + 16
(m - 4)^2 = 12 take both roots
m - 4 = ±2√3
m = ±2√3 + 4
Only the negative root makes sense
So
m = cos x = (-2√3 + 4)
Take the cosine inverse
arcos ( -2√3 + 4) = x ≈ 1.005 rads and (2pi - 1.005) ≈ 5.28 rads
So....the sum of these ≈ 6.285 rads
