What is the general solution for cos(2A)+cos(A)=1 ?

The general solution for cos(2A)+cos(A)=1 is A=0.67488…+2pin,A=2pi-0.67488…+2pin

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cos(2A)+cos(A)=1

cos (2 A) + cos (A) = 1

A = 0.67488 \dots + 2 πn, A = 2 π - 0.67488 \dots + 2 πn

Solution steps

cos (2 A) + cos (A) = 1

Subtract

1

from both sides

cos (2 A) + cos (A) - 1 = 0

Rewrite using trig identities

- 2 + cos (A) + 2 cos^{2} (A) = 0

Solve by substitution

cos (A) = \frac{- 1 + 17}{4}, cos (A) = \frac{- 1 - 17}{4}

cos (A) = \frac{- 1 + 17}{4} : A = arccos (\frac{- 1 + 17}{4}) + 2 πn, A = 2 π - arccos (\frac{- 1 + 17}{4}) + 2 πn

cos (A) = \frac{- 1 - 17}{4} :

No Solution

Combine all the solutions

A = arccos (\frac{- 1 + 17}{4}) + 2 πn, A = 2 π - arccos (\frac{- 1 + 17}{4}) + 2 πn

Show solutions in decimal form

A = 0.67488 \dots + 2 πn, A = 2 π - 0.67488 \dots + 2 πn

2 \cdot 9.19 \cdot sin (x) = 1.5406

csc^{3} (x) - 4 csc (x) = 3 cot^{2} (x)

1 + tan^{2} (θ) = 4 tan^{2} (θ)

(tan (θ) + 1) (cos (θ) + 1) (sin (θ)) = 0

6 sin (2 x) - 3 = 0

What is the general solution for cos(2A)+cos(A)=1 ?
The general solution for cos(2A)+cos(A)=1 is A=0.67488…+2pin,A=2pi-0.67488…+2pin