What is the general solution for cos^2(x)-5cos(x)-1=0 ?

The general solution for cos^2(x)-5cos(x)-1=0 is x=1.76458…+2pin,x=-1.76458…+2pin

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cos^2(x)-5cos(x)-1=0

cos^{2} (x) - 5 cos (x) - 1 = 0

x = 1.76458 \dots + 2 πn, x = - 1.76458 \dots + 2 πn

Solution steps

cos^{2} (x) - 5 cos (x) - 1 = 0

Solve by substitution

cos (x) = \frac{5 + 29}{2}, cos (x) = \frac{5 - 29}{2}

cos (x) = \frac{5 + 29}{2} :

No Solution

cos (x) = \frac{5 - 29}{2} : x = arccos (\frac{5 - 29}{2}) + 2 πn, x = - arccos (\frac{5 - 29}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{5 - 29}{2}) + 2 πn, x = - arccos (\frac{5 - 29}{2}) + 2 πn

Show solutions in decimal form

x = 1.76458 \dots + 2 πn, x = - 1.76458 \dots + 2 πn

cos (2 θ) = - cos (θ)

2 sin (x) cos (x) - 2 cos (x) - sin (x) + 1 = 0

2 cos^{2} (θ) = 1 - cos (θ)

- 5 cos^{2} (θ) = 12 sin (θ) + 4

cos^{2} (t) - 2 sin (t) - 1 = 0

What is the general solution for cos^2(x)-5cos(x)-1=0 ?
The general solution for cos^2(x)-5cos(x)-1=0 is x=1.76458…+2pin,x=-1.76458…+2pin