What is the general solution for 5sin(θ)tan(θ)-10tan(θ)+3sin(θ)-6=0 ?

The general solution for 5sin(θ)tan(θ)-10tan(θ)+3sin(θ)-6=0 is θ=-0.54041…+pin

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Popular >

5sin(θ)tan(θ)-10tan(θ)+3sin(θ)-6=0

Solution

5 sin (θ) tan (θ) - 10 tan (θ) + 3 sin (θ) - 6 = 0

Solution

θ = - 0.54041 \dots + πn

Degrees

θ = - 30.96375 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 sin (θ) tan (θ) - 10 tan (θ) + 3 sin (θ) - 6 = 0

Factor

5 sin (θ) tan (θ) - 10 tan (θ) + 3 sin (θ) - 6 : (sin (θ) - 2) (5 tan (θ) + 3)

5 sin (θ) tan (θ) - 10 tan (θ) + 3 sin (θ) - 6

= (5 sin (θ) tan (θ) - 10 tan (θ)) + (3 sin (θ) - 6)

Factor out

3

from

3 sin (θ) - 6 : 3 (sin (θ) - 2)

3 sin (θ) - 6

Rewrite

- 6

2 \cdot 3

= 3 sin (θ) + 2 \cdot 3

Factor out common term

3

= 3 (sin (θ) - 2)

Factor out

5 tan (θ)

from

5 sin (θ) tan (θ) - 10 tan (θ) : 5 tan (θ) (sin (θ) - 2)

5 sin (θ) tan (θ) - 10 tan (θ)

Rewrite

- 10

2 \cdot 5

= 5 sin (θ) tan (θ) + 2 \cdot 5 tan (θ)

Factor out common term

5 tan (θ)

= 5 tan (θ) (sin (θ) - 2)

= 3 (sin (θ) - 2) + 5 tan (θ) (sin (θ) - 2)

Factor out common term

sin (θ) - 2

= (sin (θ) - 2) (5 tan (θ) + 3)

(sin (θ) - 2) (5 tan (θ) + 3) = 0

Solving each part separately

sin (θ) - 2 = 0 or 5 tan (θ) + 3 = 0

sin (θ) - 2 = 0 :

No Solution

sin (θ) - 2 = 0

Move

2

to the right side

sin (θ) - 2 = 0

Add

2

to both sides

sin (θ) - 2 + 2 = 0 + 2

Simplify

sin (θ) = 2

sin (θ) = 2

- 1 \leq sin (x) \leq 1

No Solution

5 tan (θ) + 3 = 0 : θ = arctan (- \frac{3}{5}) + πn

5 tan (θ) + 3 = 0

Move

3

to the right side

5 tan (θ) + 3 = 0

Subtract

3

from both sides

5 tan (θ) + 3 - 3 = 0 - 3

Simplify

5 tan (θ) = - 3

5 tan (θ) = - 3

Divide both sides by

5

5 tan (θ) = - 3

Divide both sides by

5

\frac{5 tan ( θ )}{5} = \frac{- 3}{5}

Simplify

tan (θ) = - \frac{3}{5}

tan (θ) = - \frac{3}{5}

Apply trig inverse properties

tan (θ) = - \frac{3}{5}

General solutions for

tan (θ) = - \frac{3}{5}

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

θ = arctan (- \frac{3}{5}) + πn

θ = arctan (- \frac{3}{5}) + πn

Combine all the solutions

θ = arctan (- \frac{3}{5}) + πn

Show solutions in decimal form

θ = - 0.54041 \dots + πn

Popular Examples

3cos^2(θ)+6cos(θ)-4=0

3 cos^{2} (θ) + 6 cos (θ) - 4 = 0

arccos(x)=1

arccos (x) = 1

cos(x)=sqrt(3/4)

cos (x) = \frac{3}{4}

sin(x)=sin(x/2)

sin (x) = sin (\frac{x}{2})

sec(x)-2=-tan(x)-3

sec (x) - 2 = - tan (x) - 3

Frequently Asked Questions (FAQ)

What is the general solution for 5sin(θ)tan(θ)-10tan(θ)+3sin(θ)-6=0 ?
The general solution for 5sin(θ)tan(θ)-10tan(θ)+3sin(θ)-6=0 is θ=-0.54041…+pin