What is the general solution for cos(θ)= 2/(3sqrt(38)) ?

The general solution for cos(θ)= 2/(3sqrt(38)) is θ=1.46243…+2pin,θ=2pi-1.46243…+2pin

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cos(θ)= 2/(3sqrt(38))

cos (θ) = \frac{2}{3 38}

θ = 1.46243 \dots + 2 πn, θ = 2 π - 1.46243 \dots + 2 πn

Radians

θ = 1.46243 \dots + 6.28318 \dots n, θ = 4.82074 \dots + 6.28318 \dots n

Solution steps

cos (θ) = \frac{2}{3 38}

Simplify

\frac{2}{3 38} : \frac{38}{57}

\frac{2}{3 38}

Factor

38 : 219

Factor

38 = 2 \cdot 19

= 2 \cdot 19

Apply radical rule:

= 219

= \frac{2}{3 2 19}

Cancel

\frac{2}{3 2 19} : \frac{2}{3 19}

\frac{2}{3 2 19}

Apply radical rule:

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

= \frac{2}{3 \cdot 2 ^{\frac{1}{2}} 19}

Apply exponent rule:

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

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

= \frac{2 ^{1 - \frac{1}{2}}}{3 19}

Subtract the numbers:

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

= \frac{2 ^{\frac{1}{2}}}{3 19}

Apply radical rule:

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

= \frac{2}{3 19}

= \frac{2}{3 19}

Rationalize

\frac{2}{3 19} : \frac{38}{57}

\frac{2}{3 19}

Multiply by the conjugate

\frac{19}{19}

= \frac{2 19}{3 19 19}

219 = 38

219

Apply radical rule:

a b = a \cdot b

219 = 2 \cdot 19

= 2 \cdot 19

Multiply the numbers:

2 \cdot 19 = 38

= 38

31919 = 57

31919

Apply radical rule:

a a = a

1919 = 19

= 3 \cdot 19

Multiply the numbers:

3 \cdot 19 = 57

= 57

= \frac{38}{57}

= \frac{38}{57}

cos (θ) = \frac{38}{57}

Apply trig inverse properties

cos (θ) = \frac{38}{57}

General solutions for

cos (θ) = \frac{38}{57}

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

θ = arccos (\frac{38}{57}) + 2 πn, θ = 2 π - arccos (\frac{38}{57}) + 2 πn

θ = arccos (\frac{38}{57}) + 2 πn, θ = 2 π - arccos (\frac{38}{57}) + 2 πn

Show solutions in decimal form

θ = 1.46243 \dots + 2 πn, θ = 2 π - 1.46243 \dots + 2 πn

sec (x) cos (x) = sin (x)

tan (θ) - 1 - 2 = sec (θ)

3 cos (2 x) + 4 cos (4 x) - 5 = 0

cos (2 θ) = \frac{20}{29}

- 6 sin (3 x) = 0

What is the general solution for cos(θ)= 2/(3sqrt(38)) ?
The general solution for cos(θ)= 2/(3sqrt(38)) is θ=1.46243…+2pin,θ=2pi-1.46243…+2pin