What is the general solution for -3cos(pi/2-θ)=tan(θ) ?

The general solution for -3cos(pi/2-θ)=tan(θ) is θ=2pin,θ=pi+2pin,θ=1.91063…+2pin,θ=-1.91063…+2pin

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-3cos(pi/2-θ)=tan(θ)

- 3 cos (\frac{π}{2} - θ) = tan (θ)

θ = 2 πn, θ = π + 2 πn, θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

Solution steps

- 3 cos (\frac{π}{2} - θ) = tan (θ)

Rewrite using trig identities

- 3 sin (θ) = tan (θ)

Subtract

tan (θ)

from both sides

- 3 sin (θ) - tan (θ) = 0

Express with sin, cos

\frac{- sin ( θ ) - 3 cos ( θ ) sin ( θ )}{cos ( θ )} = 0

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

- sin (θ) - 3 cos (θ) sin (θ) = 0

Factor

- sin (θ) - 3 cos (θ) sin (θ) : - sin (θ) (3 cos (θ) + 1)

- sin (θ) (3 cos (θ) + 1) = 0

Solving each part separately

sin (θ) = 0 or 3 cos (θ) + 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

3 cos (θ) + 1 = 0 : θ = arccos (- \frac{1}{3}) + 2 πn, θ = - arccos (- \frac{1}{3}) + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = arccos (- \frac{1}{3}) + 2 πn, θ = - arccos (- \frac{1}{3}) + 2 πn

Show solutions in decimal form

θ = 2 πn, θ = π + 2 πn, θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

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

0.3 = sin (9 x)

sin (7 x) = cos (2 x)

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

sec^{2} (x) csc (x) - 2 csc (x) = 2 - sec^{2} (x)

What is the general solution for -3cos(pi/2-θ)=tan(θ) ?
The general solution for -3cos(pi/2-θ)=tan(θ) is θ=2pin,θ=pi+2pin,θ=1.91063…+2pin,θ=-1.91063…+2pin