What is the general solution for 5^{5-3csc(θ)}=1 ?

The general solution for 5^{5-3csc(θ)}=1 is θ=0.64350…+2pin,θ=pi-0.64350…+2pin

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Popular >

5^{5-3csc(θ)}=1

Solution

5^{5 - 3 c s c (θ)} = 1

Solution

θ = 0.64350 \dots + 2 πn, θ = π - 0.64350 \dots + 2 πn

Degrees

θ = 36.86989 \dots^{\circ} + 36 0^{\circ} n, θ = 143.13010 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5^{5 - 3 c s c (θ)} = 1

Apply exponent rules

5^{5 - 3 c s c (θ)} = 1

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (5^{5 - 3 c s c (θ)}) = ln (1)

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (5^{5 - 3 c s c (θ)}) = (5 - 3 csc (θ)) ln (5)

(5 - 3 csc (θ)) ln (5) = ln (1)

(5 - 3 csc (θ)) ln (5) = ln (1)

Solve

(5 - 3 csc (θ)) ln (5) = ln (1) : csc (θ) = \frac{5}{3}

(5 - 3 csc (θ)) ln (5) = ln (1)

Divide both sides by

ln (5)

(5 - 3 csc (θ)) ln (5) = ln (1)

Divide both sides by

ln (5)

\frac{( 5 - 3 csc ( θ ) ) ln ( 5 )}{ln ( 5 )} = \frac{ln ( 1 )}{ln ( 5 )}

Simplify

\frac{( 5 - 3 csc ( θ ) ) ln ( 5 )}{ln ( 5 )} = \frac{ln ( 1 )}{ln ( 5 )}

Simplify

\frac{( 5 - 3 csc ( θ ) ) ln ( 5 )}{ln ( 5 )} : 5 - 3 csc (θ)

\frac{( 5 - 3 csc ( θ ) ) ln ( 5 )}{ln ( 5 )}

Cancel the common factor:

ln (5)

= 5 - 3 csc (θ)

Simplify

\frac{ln ( 1 )}{ln ( 5 )} : 0

\frac{ln ( 1 )}{ln ( 5 )}

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

= \frac{0}{ln ( 5 )}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

5 - 3 csc (θ) = 0

5 - 3 csc (θ) = 0

5 - 3 csc (θ) = 0

Move

5

to the right side

5 - 3 csc (θ) = 0

Subtract

5

from both sides

5 - 3 csc (θ) - 5 = 0 - 5

Simplify

- 3 csc (θ) = - 5

- 3 csc (θ) = - 5

Divide both sides by

- 3

- 3 csc (θ) = - 5

Divide both sides by

- 3

\frac{- 3 csc ( θ )}{- 3} = \frac{- 5}{- 3}

Simplify

csc (θ) = \frac{5}{3}

csc (θ) = \frac{5}{3}

csc (θ) = \frac{5}{3}

Apply trig inverse properties

csc (θ) = \frac{5}{3}

General solutions for

csc (θ) = \frac{5}{3}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

θ = arccsc (\frac{5}{3}) + 2 πn, θ = π - arccsc (\frac{5}{3}) + 2 πn

θ = arccsc (\frac{5}{3}) + 2 πn, θ = π - arccsc (\frac{5}{3}) + 2 πn

Show solutions in decimal form

θ = 0.64350 \dots + 2 πn, θ = π - 0.64350 \dots + 2 πn

Popular Examples

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sin (z) = \frac{4}{7}

2cos(θ)-2cos(3θ)=0

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5-7sin(A)-2cos^2(A)=0

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solvefor x,ln(y)+y^2=sin(x)+c solve for

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5/2 =2+cos(x+(2pi)/3)

\frac{5}{2} = 2 + cos (x + \frac{2 π}{3})

Frequently Asked Questions (FAQ)

What is the general solution for 5^{5-3csc(θ)}=1 ?
The general solution for 5^{5-3csc(θ)}=1 is θ=0.64350…+2pin,θ=pi-0.64350…+2pin