What is the general solution for cos(θ)=(-8)/(sqrt(14)*\sqrt{20)} ?

The general solution for cos(θ)=(-8)/(sqrt(14)*\sqrt{20)} is θ=2.06927…+2pin,θ=-2.06927…+2pin

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cos(θ)=(-8)/(sqrt(14)*\sqrt{20)}

Solution

cos (θ) = \frac{- 8}{14 \cdot 20}

Solution

θ = 2.06927 \dots + 2 πn, θ = - 2.06927 \dots + 2 πn

Radians

θ = 2.06927 \dots + 6.28318 \dots n, θ = - 2.06927 \dots + 6.28318 \dots n

Solution steps

cos (θ) = \frac{- 8}{14 20}

Simplify

\frac{- 8}{14 20} : - \frac{2 70}{35}

\frac{- 8}{14 20}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{14 20}

Cancel

\frac{8}{14 20} : \frac{2 ^{2} 2}{2 7 5}

\frac{8}{14 20}

Factor

8 : 2^{3}

Factor

8 = 2^{3}

Factor

14 : 27

Factor

14 = 2 \cdot 7

= 2 \cdot 7

Apply radical rule:

n ab = n a n b

= 27

Factor

20 : 25

Factor

20 = 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

n ab = n a n b

= 5 2^{2}

Simplify

2^{2} : 2

2^{2}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 2

= 25

= \frac{2 ^{3}}{2 2 7 5}

Cancel

\frac{2 ^{3}}{2 7 \cdot 2 5} : \frac{2 ^{\frac{5}{2}}}{2 7 5}

\frac{2 ^{3}}{2 7 \cdot 2 5}

Apply radical rule:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{3}}{2 \cdot 2 ^{\frac{1}{2}} 7 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{3}}{2 ^{\frac{1}{2}}} = 2^{3 - \frac{1}{2}}

= \frac{2 ^{3 - \frac{1}{2}}}{2 7 5}

Subtract the numbers:

3 - \frac{1}{2} = \frac{5}{2}

= \frac{2 ^{\frac{5}{2}}}{2 7 5}

= \frac{2 ^{\frac{5}{2}}}{2 7 5}

2^{\frac{5}{2}} = 2^{2} 2

2^{\frac{5}{2}}

2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}}

= 2^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{2}}

Refine

= 2^{2} 2

= \frac{2 ^{2} 2}{2 7 5}

= - \frac{2 ^{2} 2}{2 7 5}

Simplify

275 : 235

275

Apply radical rule:

a b = a \cdot b

75 = 7 \cdot 5

= 2 7 \cdot 5

Multiply the numbers:

7 \cdot 5 = 35

= 235

= - \frac{2 ^{2} 2}{2 35}

Cancel the common factor:

2

= - \frac{2 2}{35}

Rationalize

- \frac{2 2}{35} : - \frac{2 70}{35}

- \frac{2 2}{35}

Multiply by the conjugate

\frac{35}{35}

= - \frac{2 2 35}{35 35}

2235 = 270

2235

Apply radical rule:

a b = a \cdot b

235 = 2 \cdot 35

= 2 2 \cdot 35

Multiply the numbers:

2 \cdot 35 = 70

= 270

3535 = 35

3535

Apply radical rule:

a a = a

3535 = 35

= 35

= - \frac{2 70}{35}

= - \frac{2 70}{35}

cos (θ) = - \frac{2 70}{35}

Apply trig inverse properties

cos (θ) = - \frac{2 70}{35}

General solutions for

cos (θ) = - \frac{2 70}{35}

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

θ = arccos (- \frac{2 70}{35}) + 2 πn, θ = - arccos (- \frac{2 70}{35}) + 2 πn

θ = arccos (- \frac{2 70}{35}) + 2 πn, θ = - arccos (- \frac{2 70}{35}) + 2 πn

Show solutions in decimal form

θ = 2.06927 \dots + 2 πn, θ = - 2.06927 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ)=(-8)/(sqrt(14)*\sqrt{20)} ?
The general solution for cos(θ)=(-8)/(sqrt(14)*\sqrt{20)} is θ=2.06927…+2pin,θ=-2.06927…+2pin