What is the general solution for cos(θ)=(-2)/((sqrt(30)+(7))) ?

The general solution for cos(θ)=(-2)/((sqrt(30)+(7))) is θ=1.73178…+2pin,θ=-1.73178…+2pin

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cos(θ)=(-2)/((sqrt(30)+(7)))

cos (θ) = \frac{- 2}{( 30 + ( 7 ) )}

θ = 1.73178 \dots + 2 πn, θ = - 1.73178 \dots + 2 πn

Radians

θ = 1.73178 \dots + 6.28318 \dots n, θ = - 1.73178 \dots + 6.28318 \dots n

Solution steps

cos (θ) = \frac{- 2}{( 30 + ( 7 ) )}

Simplify

\frac{- 2}{30 + 7} : \frac{2 ( 30 - 7 )}{19}

\frac{- 2}{30 + 7}

Apply the fraction rule:

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

= - \frac{2}{30 + 7}

Rationalize

- \frac{2}{7 + 30} : \frac{2 ( 30 - 7 )}{19}

- \frac{2}{7 + 30}

Multiply by the conjugate

\frac{30 - 7}{30 - 7}

= - \frac{2 ( 30 - 7 )}{( 30 + 7 ) ( 30 - 7 )}

(30 + 7) (30 - 7) = - 19

(30 + 7) (30 - 7)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 30, b = 7

= (30)^{2} - 7^{2}

Simplify

(30)^{2} - 7^{2} : - 19

(30)^{2} - 7^{2}

(30)^{2} = 30

(30)^{2}

Apply radical rule:

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

= (3 0^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3 0^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 30

7^{2} = 49

7^{2}

7^{2} = 49

= 49

= 30 - 49

Subtract the numbers:

30 - 49 = - 19

= - 19

= - 19

= - \frac{2 ( 30 - 7 )}{- 19}

Apply the fraction rule:

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

= - (- \frac{2 ( 30 - 7 )}{19})

Apply rule

- (- a) = a

= \frac{2 ( 30 - 7 )}{19}

= \frac{2 ( 30 - 7 )}{19}

cos (θ) = \frac{2 ( 30 - 7 )}{19}

Apply trig inverse properties

cos (θ) = \frac{2 ( 30 - 7 )}{19}

General solutions for

cos (θ) = \frac{2 ( 30 - 7 )}{19}

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

θ = arccos (\frac{2 ( 30 - 7 )}{19}) + 2 πn, θ = - arccos (\frac{2 ( 30 - 7 )}{19}) + 2 πn

θ = arccos (\frac{2 ( 30 - 7 )}{19}) + 2 πn, θ = - arccos (\frac{2 ( 30 - 7 )}{19}) + 2 πn

Show solutions in decimal form

θ = 1.73178 \dots + 2 πn, θ = - 1.73178 \dots + 2 πn

cos^{3} (x) - 2 sin (x) - 0.7 = 0

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

2 - 3 sin (x) = 0

y = cos (x), x

\frac{cos ( θ )}{sin ( θ )} = 0

What is the general solution for cos(θ)=(-2)/((sqrt(30)+(7))) ?
The general solution for cos(θ)=(-2)/((sqrt(30)+(7))) is θ=1.73178…+2pin,θ=-1.73178…+2pin