What is the general solution for 4^{1/3}sec(x)=4(1/(4^{1/3)})csc(x) ?

The general solution for 4^{1/3}sec(x)=4(1/(4^{1/3)})csc(x) is x=1.00863…+pin

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4^{1/3}sec(x)=4(1/(4^{1/3)})csc(x)

4^{\frac{1}{3}} sec (x) = 4 (\frac{1}{4 ^{\frac{1}{3}}}) csc (x)

x = 1.00863 \dots + πn

Solution steps

4^{\frac{1}{3}} sec (x) = 4 (\frac{1}{4 ^{\frac{1}{3}}}) csc (x)

Subtract

4 \frac{1}{4 ^{\frac{1}{3}}} csc (x)

from both sides

4^{\frac{1}{3}} sec (x) - 4^{\frac{2}{3}} csc (x) = 0

Express with sin, cos

\frac{4 ^{\frac{1}{3}} sin ( x ) - 4 ^{\frac{2}{3}} cos ( x )}{cos ( x ) sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4^{\frac{1}{3}} sin (x) - 4^{\frac{2}{3}} cos (x) = 0

Rewrite using trig identities

4^{\frac{1}{3}} tan (x) - 4^{\frac{2}{3}} = 0

Move

4^{\frac{2}{3}}

to the right side

4^{\frac{1}{3}} tan (x) = 4^{\frac{2}{3}}

Divide both sides by

4^{\frac{1}{3}}

tan (x) = 4^{\frac{1}{3}}

Apply trig inverse properties

x = arctan (4^{\frac{1}{3}}) + πn

Show solutions in decimal form

x = 1.00863 \dots + πn

\frac{1 + tan ^{2} ( x )}{1 + tan ^{2} ( x )} = 1 - sin^{2} (x)

θ, tan (θ) = \frac{2 3}{2}

\frac{- 3 + 4 cos ^{2} ( θ )}{1 - 2 sin ( θ )} = a + b sin (θ)

2 sin^{2} (x) - 7 sin (x) = - 3, 0 \leq x \leq 2 π

sin^{2} (x) + cos^{2} (x) = cos (2 x)

What is the general solution for 4^{1/3}sec(x)=4(1/(4^{1/3)})csc(x) ?
The general solution for 4^{1/3}sec(x)=4(1/(4^{1/3)})csc(x) is x=1.00863…+pin