What is the general solution for (3tan^2(x)-5)/(cos(x)+1)=0 ?

The general solution for (3tan^2(x)-5)/(cos(x)+1)=0 is x=0.91173…+pin,x=-0.91173…+pin

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(3tan^2(x)-5)/(cos(x)+1)=0

\frac{3 tan ^{2} ( x ) - 5}{cos ( x ) + 1} = 0

x = 0.91173 \dots + πn, x = - 0.91173 \dots + πn

Degrees

x = 52.23875 \dots^{\circ} + 18 0^{\circ} n, x = - 52.23875 \dots^{\circ} + 18 0^{\circ} n

Solution steps

\frac{3 tan ^{2} ( x ) - 5}{cos ( x ) + 1} = 0

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

3 tan^{2} (x) - 5 = 0

Solve by substitution

3 tan^{2} (x) - 5 = 0

Let:

tan (x) = u

3 u^{2} - 5 = 0

3 u^{2} - 5 = 0 : u = \frac{5}{3}, u = - \frac{5}{3}

3 u^{2} - 5 = 0

Move

5

to the right side

3 u^{2} - 5 = 0

Add

5

to both sides

3 u^{2} - 5 + 5 = 0 + 5

Simplify

3 u^{2} = 5

3 u^{2} = 5

Divide both sides by

3

3 u^{2} = 5

Divide both sides by

3

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{5}{3}, u = - \frac{5}{3}

Substitute back

u = tan (x)

tan (x) = \frac{5}{3}, tan (x) = - \frac{5}{3}

tan (x) = \frac{5}{3}, tan (x) = - \frac{5}{3}

tan (x) = \frac{5}{3} : x = arctan (\frac{5}{3}) + πn

tan (x) = \frac{5}{3}

Apply trig inverse properties

tan (x) = \frac{5}{3}

General solutions for

tan (x) = \frac{5}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{5}{3}) + πn

x = arctan (\frac{5}{3}) + πn

tan (x) = - \frac{5}{3} : x = arctan (- \frac{5}{3}) + πn

tan (x) = - \frac{5}{3}

Apply trig inverse properties

tan (x) = - \frac{5}{3}

General solutions for

tan (x) = - \frac{5}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{5}{3}) + πn

x = arctan (- \frac{5}{3}) + πn

Combine all the solutions

x = arctan (\frac{5}{3}) + πn, x = arctan (- \frac{5}{3}) + πn

Show solutions in decimal form

x = 0.91173 \dots + πn, x = - 0.91173 \dots + πn

sin (3 x) = \frac{1}{2}, 0 \leq x \leq 2 π

cos (θ) = sec (θ), 0 \leq θ < 2 π

\frac{441}{841} + sin^{2} (t) = 1

3 cot (\frac{2 x}{3}) - 1 = 0

2 sin (3 x) = 1, 0 \leq x \leq 4 π

What is the general solution for (3tan^2(x)-5)/(cos(x)+1)=0 ?
The general solution for (3tan^2(x)-5)/(cos(x)+1)=0 is x=0.91173…+pin,x=-0.91173…+pin