What is the general solution for cos(2x)=0.685 ?

The general solution for cos(2x)=0.685 is x=(0.81619…)/2+pin,x=pi-(0.81619…)/2+pin

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cos(2x)=0.685

cos (2 x) = 0.685

x = \frac{0.81619 \dots}{2} + πn, x = π - \frac{0.81619 \dots}{2} + πn

Degrees

x = 23.38219 \dots^{\circ} + 18 0^{\circ} n, x = 156.61780 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cos (2 x) = 0.685

Apply trig inverse properties

cos (2 x) = 0.685

General solutions for

cos (2 x) = 0.685

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x = arccos (0.685) + 2 πn, 2 x = 2 π - arccos (0.685) + 2 πn

2 x = arccos (0.685) + 2 πn, 2 x = 2 π - arccos (0.685) + 2 πn

Solve

2 x = arccos (0.685) + 2 πn : x = \frac{arccos ( 0.685 )}{2} + πn

2 x = arccos (0.685) + 2 πn

Divide both sides by

2

2 x = arccos (0.685) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( 0.685 )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( 0.685 )}{2} + πn

x = \frac{arccos ( 0.685 )}{2} + πn

Solve

2 x = 2 π - arccos (0.685) + 2 πn : x = π - \frac{arccos ( 0.685 )}{2} + πn

2 x = 2 π - arccos (0.685) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (0.685) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 0.685 )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( 0.685 )}{2} + πn

x = π - \frac{arccos ( 0.685 )}{2} + πn

x = \frac{arccos ( 0.685 )}{2} + πn, x = π - \frac{arccos ( 0.685 )}{2} + πn

Show solutions in decimal form

x = \frac{0.81619 \dots}{2} + πn, x = π - \frac{0.81619 \dots}{2} + πn

4 tan^{2} (x) = 12

cos (θ) = - 0.71

cos (θ) = \frac{60}{61}

cos (\frac{1}{2} x) = 0

\frac{4 cos ( θ ) - 1}{tan ( θ )} = 2 sin (θ)

What is the general solution for cos(2x)=0.685 ?
The general solution for cos(2x)=0.685 is x=(0.81619…)/2+pin,x=pi-(0.81619…)/2+pin