What is the general solution for cos(2x)+5cos(x)=3 ?

The general solution for cos(2x)+5cos(x)=3 is x=0.87960…+2pin,x=2pi-0.87960…+2pin

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cos(2x)+5cos(x)=3

cos (2 x) + 5 cos (x) = 3

x = 0.87960 \dots + 2 πn, x = 2 π - 0.87960 \dots + 2 πn

Solution steps

cos (2 x) + 5 cos (x) = 3

Subtract

3

from both sides

cos (2 x) + 5 cos (x) - 3 = 0

Rewrite using trig identities

- 4 + 2 cos^{2} (x) + 5 cos (x) = 0

Solve by substitution

cos (x) = \frac{- 5 + 57}{4}, cos (x) = \frac{- 5 - 57}{4}

cos (x) = \frac{- 5 + 57}{4} : x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

cos (x) = \frac{- 5 - 57}{4} :

No Solution

Combine all the solutions

x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

Show solutions in decimal form

x = 0.87960 \dots + 2 πn, x = 2 π - 0.87960 \dots + 2 πn

sin (θ) = \frac{207.91 \cdot sin ( 31.7697 7 ^{\circ} )}{238.99}

15 = - 40 sin (18 (x - 10))

tan^{2} (x) - tan (x) - 6 = 0

\frac{1}{sec ( 2 x ) - 1} - \frac{1}{sec ( 2 x ) + 1} = 6

sin (2 π + x) - sin (2 π - x) = - 1

What is the general solution for cos(2x)+5cos(x)=3 ?
The general solution for cos(2x)+5cos(x)=3 is x=0.87960…+2pin,x=2pi-0.87960…+2pin