What is the general solution for cot(θ)+2csc(θ)=4,0<= θ<= 360 ?

The general solution for cot(θ)+2csc(θ)=4,0<= θ<= 360 is θ=0.75142…,θ=2.88012…

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cot(θ)+2csc(θ)=4,0<= θ<= 360

cot (θ) + 2 csc (θ) = 4, 0^{\circ} \leq θ \leq 36 0^{\circ}

θ = 0.75142 \dots, θ = 2.88012 \dots

Solution steps

cot (θ) + 2 csc (θ) = 4, 0^{\circ} \leq θ \leq 36 0^{\circ}

Subtract

4

from both sides

cot (θ) + 2 csc (θ) - 4 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + 2 \cdot \frac{1}{sin ( θ )} - 4 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + 2 \cdot \frac{1}{sin ( θ )} - 4 : \frac{cos ( θ ) + 2 - 4 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 2 - 4 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 2 - 4 sin (θ) = 0

Add

4 sin (θ)

to both sides

cos (θ) + 2 = 4 sin (θ)

Square both sides

(cos (θ) + 2)^{2} = (4 sin (θ))^{2}

Subtract

(4 sin (θ))^{2}

from both sides

(cos (θ) + 2)^{2} - 16 sin^{2} (θ) = 0

Rewrite using trig identities

- 12 + 17 cos^{2} (θ) + 4 cos (θ) = 0

Solve by substitution

cos (θ) = \frac{2 ( 2 13 - 1 )}{17}, cos (θ) = - \frac{2 ( 1 + 2 13 )}{17}

cos (θ) = \frac{2 ( 2 13 - 1 )}{17}, 0 \leq θ \leq 36 0^{\circ} : θ = arccos (\frac{2 ( 2 13 - 1 )}{17}), θ = 36 0^{\circ} - arccos (\frac{2 ( 2 13 - 1 )}{17})

cos (θ) = - \frac{2 ( 1 + 2 13 )}{17}, 0 \leq θ \leq 36 0^{\circ} : θ = arccos (- \frac{2 ( 1 + 2 13 )}{17}), θ = - arccos (- \frac{2 ( 1 + 2 13 )}{17}) + 36 0^{\circ}

Combine all the solutions

θ = arccos (\frac{2 ( 2 13 - 1 )}{17}), θ = 36 0^{\circ} - arccos (\frac{2 ( 2 13 - 1 )}{17}), θ = arccos (- \frac{2 ( 1 + 2 13 )}{17}), θ = - arccos (- \frac{2 ( 1 + 2 13 )}{17}) + 36 0^{\circ}

Verify solutions by plugging them into the original equation

θ = arccos (\frac{2 ( 2 13 - 1 )}{17}), θ = arccos (- \frac{2 ( 1 + 2 13 )}{17})

Show solutions in decimal form

θ = 0.75142 \dots, θ = 2.88012 \dots

4 sin (x) = 23

sin (x) = 180

2 sin^{2} (x) - 3 cos (- x) - 3 = 0

49.55 cos (θ) - 30 sin (θ) = 1.225

tan (2 x) + sin (2 x) + cos (2 x) = sec (2 x)

What is the general solution for cot(θ)+2csc(θ)=4,0<= θ<= 360 ?
The general solution for cot(θ)+2csc(θ)=4,0<= θ<= 360 is θ=0.75142…,θ=2.88012…