What is the general solution for 23cos(x)=-17-cos(x) ?

The general solution for 23cos(x)=-17-cos(x) is x=2.35793…+2pin,x=-2.35793…+2pin

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23cos(x)=-17-cos(x)

23 cos (x) = - 17 - cos (x)

x = 2.35793 \dots + 2 πn, x = - 2.35793 \dots + 2 πn

Degrees

x = 135.09947 \dots^{\circ} + 36 0^{\circ} n, x = - 135.09947 \dots^{\circ} + 36 0^{\circ} n

Solution steps

23 cos (x) = - 17 - cos (x)

Solve by substitution

23 cos (x) = - 17 - cos (x)

Let:

cos (x) = u

23 u = - 17 - u

23 u = - 17 - u : u = - \frac{17}{24}

23 u = - 17 - u

Move

u

to the left side

23 u = - 17 - u

Add

u

to both sides

23 u + u = - 17 - u + u

Simplify

24 u = - 17

24 u = - 17

Divide both sides by

24

24 u = - 17

Divide both sides by

24

\frac{24 u}{24} = \frac{- 17}{24}

Simplify

u = - \frac{17}{24}

u = - \frac{17}{24}

Substitute back

u = cos (x)

cos (x) = - \frac{17}{24}

cos (x) = - \frac{17}{24}

cos (x) = - \frac{17}{24} : x = arccos (- \frac{17}{24}) + 2 πn, x = - arccos (- \frac{17}{24}) + 2 πn

cos (x) = - \frac{17}{24}

Apply trig inverse properties

cos (x) = - \frac{17}{24}

General solutions for

cos (x) = - \frac{17}{24}

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

x = arccos (- \frac{17}{24}) + 2 πn, x = - arccos (- \frac{17}{24}) + 2 πn

x = arccos (- \frac{17}{24}) + 2 πn, x = - arccos (- \frac{17}{24}) + 2 πn

Combine all the solutions

x = arccos (- \frac{17}{24}) + 2 πn, x = - arccos (- \frac{17}{24}) + 2 πn

Show solutions in decimal form

x = 2.35793 \dots + 2 πn, x = - 2.35793 \dots + 2 πn

8 cos^{2} (θ) + 14 cos (θ) - 11 = 8 cos (θ) - 6

tan (x) cos (x) + cos (x) = 0

sin^{2} (x) (tan^{2} (x)) = 1 - sin^{2} (x)

2 sin^{2} (x) = 3 - 5 cos (x)

- 3 cos (2 θ) - 18 sin (θ) - 8 = - 5 sin (θ) - 6

What is the general solution for 23cos(x)=-17-cos(x) ?
The general solution for 23cos(x)=-17-cos(x) is x=2.35793…+2pin,x=-2.35793…+2pin