What is the general solution for 7tan(θ)sin(θ)-6tan(θ)=0 ?

The general solution for 7tan(θ)sin(θ)-6tan(θ)=0 is θ=pin,θ=1.02969…+2pin,θ=pi-1.02969…+2pin

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7tan(θ)sin(θ)-6tan(θ)=0

7 tan (θ) sin (θ) - 6 tan (θ) = 0

θ = πn, θ = 1.02969 \dots + 2 πn, θ = π - 1.02969 \dots + 2 πn

Degrees

θ = 0^{\circ} + 18 0^{\circ} n, θ = 58.99728 \dots^{\circ} + 36 0^{\circ} n, θ = 121.00271 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 tan (θ) sin (θ) - 6 tan (θ) = 0

Factor

7 tan (θ) sin (θ) - 6 tan (θ) : tan (θ) (7 sin (θ) - 6)

7 tan (θ) sin (θ) - 6 tan (θ)

Factor out common term

tan (θ)

= tan (θ) (7 sin (θ) - 6)

tan (θ) (7 sin (θ) - 6) = 0

Solving each part separately

tan (θ) = 0 or 7 sin (θ) - 6 = 0

tan (θ) = 0 : θ = πn

tan (θ) = 0

General solutions for

tan (θ) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

7 sin (θ) - 6 = 0 : θ = arcsin (\frac{6}{7}) + 2 πn, θ = π - arcsin (\frac{6}{7}) + 2 πn

7 sin (θ) - 6 = 0

Move

6

to the right side

7 sin (θ) - 6 = 0

Add

6

to both sides

7 sin (θ) - 6 + 6 = 0 + 6

Simplify

7 sin (θ) = 6

7 sin (θ) = 6

Divide both sides by

7

7 sin (θ) = 6

Divide both sides by

7

\frac{7 sin ( θ )}{7} = \frac{6}{7}

Simplify

sin (θ) = \frac{6}{7}

sin (θ) = \frac{6}{7}

Apply trig inverse properties

sin (θ) = \frac{6}{7}

General solutions for

sin (θ) = \frac{6}{7}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

θ = arcsin (\frac{6}{7}) + 2 πn, θ = π - arcsin (\frac{6}{7}) + 2 πn

θ = arcsin (\frac{6}{7}) + 2 πn, θ = π - arcsin (\frac{6}{7}) + 2 πn

Combine all the solutions

θ = πn, θ = arcsin (\frac{6}{7}) + 2 πn, θ = π - arcsin (\frac{6}{7}) + 2 πn

Show solutions in decimal form

θ = πn, θ = 1.02969 \dots + 2 πn, θ = π - 1.02969 \dots + 2 πn

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

5 cos (θ) = 0

sin (2 θ) = 4 cos (θ)

tan (θ) = \frac{4}{4}

tan (θ) = \frac{4}{6}

What is the general solution for 7tan(θ)sin(θ)-6tan(θ)=0 ?
The general solution for 7tan(θ)sin(θ)-6tan(θ)=0 is θ=pin,θ=1.02969…+2pin,θ=pi-1.02969…+2pin