What is the general solution for 7cot(z)+tan(z)=7csc(z) ?

The general solution for 7cot(z)+tan(z)=7csc(z) is z=1.40334…+2pin,z=2pi-1.40334…+2pin

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7cot(z)+tan(z)=7csc(z)

7 cot (z) + tan (z) = 7 csc (z)

z = 1.40334 \dots + 2 πn, z = 2 π - 1.40334 \dots + 2 πn

Solution steps

7 cot (z) + tan (z) = 7 csc (z)

Subtract

7 csc (z)

from both sides

7 cot (z) + tan (z) - 7 csc (z) = 0

Express with sin, cos

\frac{sin ^{2} ( z ) + ( - 7 + 7 cos ( z ) ) cos ( z )}{cos ( z ) sin ( z )} = 0

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

sin^{2} (z) + (- 7 + 7 cos (z)) cos (z) = 0

Rewrite using trig identities

1 + 6 cos^{2} (z) - 7 cos (z) = 0

Solve by substitution

cos (z) = 1, cos (z) = \frac{1}{6}

cos (z) = 1 : z = 2 πn

cos (z) = \frac{1}{6} : z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Combine all the solutions

z = 2 πn, z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Since the equation is undefined for:

2 πn

z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Show solutions in decimal form

z = 1.40334 \dots + 2 πn, z = 2 π - 1.40334 \dots + 2 πn

3 + sin (2 x) = 1 - 6 sin (2 x)

cos (x) = 0, sin (x) = cos (x), sin (x) = 0

2 a sin (x) + a = 0

tan (θ) = (\frac{2}{3})

tan (θ) = 8.1443

What is the general solution for 7cot(z)+tan(z)=7csc(z) ?
The general solution for 7cot(z)+tan(z)=7csc(z) is z=1.40334…+2pin,z=2pi-1.40334…+2pin