What is the general solution for (1+tan(x))(5sin(x)-2)=0 ?

The general solution for (1+tan(x))(5sin(x)-2)=0 is x=(3pi)/4+pin,x=0.41151…+2pin,x=pi-0.41151…+2pin

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Popular >

(1+tan(x))(5sin(x)-2)=0

Solution

(1 + tan (x)) (5 sin (x) - 2) = 0

Solution

x = \frac{3 π}{4} + πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 156.42182 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(1 + tan (x)) (5 sin (x) - 2) = 0

Solving each part separately

1 + tan (x) = 0 or 5 sin (x) - 2 = 0

1 + tan (x) = 0 : x = \frac{3 π}{4} + πn

1 + tan (x) = 0

Move

1

to the right side

1 + tan (x) = 0

Subtract

1

from both sides

1 + tan (x) - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

5 sin (x) - 2 = 0 : x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

5 sin (x) - 2 = 0

Move

2

to the right side

5 sin (x) - 2 = 0

Add

2

to both sides

5 sin (x) - 2 + 2 = 0 + 2

Simplify

5 sin (x) = 2

5 sin (x) = 2

Divide both sides by

5

5 sin (x) = 2

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{2}{5}

Simplify

sin (x) = \frac{2}{5}

sin (x) = \frac{2}{5}

Apply trig inverse properties

sin (x) = \frac{2}{5}

General solutions for

sin (x) = \frac{2}{5}

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

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

Show solutions in decimal form

x = \frac{3 π}{4} + πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (1+tan(x))(5sin(x)-2)=0 ?
The general solution for (1+tan(x))(5sin(x)-2)=0 is x=(3pi)/4+pin,x=0.41151…+2pin,x=pi-0.41151…+2pin