What is the general solution for cos^2(θ)+0.339cos(θ)-1=0 ?

The general solution for cos^2(θ)+0.339cos(θ)-1=0 is θ=0.56467…+2pin,θ=2pi-0.56467…+2pin

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Popular >

cos^2(θ)+0.339cos(θ)-1=0

Solution

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

Solution

θ = 0.56467 \dots + 2 πn, θ = 2 π - 0.56467 \dots + 2 πn

Degrees

θ = 32.35340 \dots^{\circ} + 36 0^{\circ} n, θ = 327.64659 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

Solve by substitution

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

Let:

cos (θ) = u

u^{2} + 0.339 u - 1 = 0

u^{2} + 0.339 u - 1 = 0 : u = \frac{- 339 + 4114921}{2000}, u = \frac{- 339 - 4114921}{2000}

u^{2} + 0.339 u - 1 = 0

Multiply both sides by

1000

u^{2} + 0.339 u - 1 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

3

digits to the right of the decimal point, therefore multiply by

1000

u^{2} \cdot 1000 + 0.339 u \cdot 1000 - 1 \cdot 1000 = 0 \cdot 1000

Refine

1000 u^{2} + 339 u - 1000 = 0

1000 u^{2} + 339 u - 1000 = 0

Solve with the quadratic formula

1000 u^{2} + 339 u - 1000 = 0

Quadratic Equation Formula:

For

a = 1000, b = 339, c = - 1000

u_{1, 2} = \frac{- 339 \pm 33 9 ^{2} - 4 \cdot 1000 ( - 1000 )}{2 \cdot 1000}

u_{1, 2} = \frac{- 339 \pm 33 9 ^{2} - 4 \cdot 1000 ( - 1000 )}{2 \cdot 1000}

33 9^{2} - 4 \cdot 1000 (- 1000) = 4114921

33 9^{2} - 4 \cdot 1000 (- 1000)

Apply rule

- (- a) = a

= 33 9^{2} + 4 \cdot 1000 \cdot 1000

Multiply the numbers:

4 \cdot 1000 \cdot 1000 = 4000000

= 33 9^{2} + 4000000

33 9^{2} = 114921

= 114921 + 4000000

Add the numbers:

114921 + 4000000 = 4114921

= 4114921

u_{1, 2} = \frac{- 339 \pm 4114921}{2 \cdot 1000}

Separate the solutions

u_{1} = \frac{- 339 + 4114921}{2 \cdot 1000}, u_{2} = \frac{- 339 - 4114921}{2 \cdot 1000}

u = \frac{- 339 + 4114921}{2 \cdot 1000} : \frac{- 339 + 4114921}{2000}

\frac{- 339 + 4114921}{2 \cdot 1000}

Multiply the numbers:

2 \cdot 1000 = 2000

= \frac{- 339 + 4114921}{2000}

u = \frac{- 339 - 4114921}{2 \cdot 1000} : \frac{- 339 - 4114921}{2000}

\frac{- 339 - 4114921}{2 \cdot 1000}

Multiply the numbers:

2 \cdot 1000 = 2000

= \frac{- 339 - 4114921}{2000}

The solutions to the quadratic equation are:

u = \frac{- 339 + 4114921}{2000}, u = \frac{- 339 - 4114921}{2000}

Substitute back

u = cos (θ)

cos (θ) = \frac{- 339 + 4114921}{2000}, cos (θ) = \frac{- 339 - 4114921}{2000}

cos (θ) = \frac{- 339 + 4114921}{2000}, cos (θ) = \frac{- 339 - 4114921}{2000}

cos (θ) = \frac{- 339 + 4114921}{2000} : θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

cos (θ) = \frac{- 339 + 4114921}{2000}

Apply trig inverse properties

cos (θ) = \frac{- 339 + 4114921}{2000}

General solutions for

cos (θ) = \frac{- 339 + 4114921}{2000}

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

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

cos (θ) = \frac{- 339 - 4114921}{2000} :

No Solution

cos (θ) = \frac{- 339 - 4114921}{2000}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

Show solutions in decimal form

θ = 0.56467 \dots + 2 πn, θ = 2 π - 0.56467 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(θ)+0.339cos(θ)-1=0 ?
The general solution for cos^2(θ)+0.339cos(θ)-1=0 is θ=0.56467…+2pin,θ=2pi-0.56467…+2pin