What is the general solution for 6tan(x)-5csc(x)=0 ?

The general solution for 6tan(x)-5csc(x)=0 is x=0.84106…+2pin,x=2pi-0.84106…+2pin

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6tan(x)-5csc(x)=0

6 tan (x) - 5 csc (x) = 0

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Solution steps

6 tan (x) - 5 csc (x) = 0

Express with sin, cos

\frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

- 5 cos (x) + 6 sin^{2} (x) = 0

Rewrite using trig identities

(1 - cos^{2} (x)) \cdot 6 - 5 cos (x) = 0

Solve by substitution

cos (x) = - \frac{3}{2}, cos (x) = \frac{2}{3}

cos (x) = - \frac{3}{2} :

No Solution

cos (x) = \frac{2}{3} : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Combine all the solutions

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

2 (19) tan (\frac{θ}{2}) = 6.3

3 csc (x) - 6 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

tan (\frac{x}{2} + \frac{π}{3}) = - 1

- sec^{2} (x) + 1 = - 5 sec (x) - 5

sec (t) = \frac{1}{- \frac{2}{2}}

What is the general solution for 6tan(x)-5csc(x)=0 ?
The general solution for 6tan(x)-5csc(x)=0 is x=0.84106…+2pin,x=2pi-0.84106…+2pin