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Popular >

tan(x)csc(x)-2tan(x)=3csc(x)-6

Solution

tan (x) csc (x) - 2 tan (x) = 3 csc (x) - 6

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 1.24904 \dots + πn

+1

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) csc (x) - 2 tan (x) = 3 csc (x) - 6

Subtract

3 csc (x) - 6

from both sides

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6 = 0

Factor

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6 : (csc (x) - 2) (tan (x) - 3)

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6

= (tan (x) csc (x) - 2 tan (x)) + (- 3 csc (x) + 6)

Factor out

- 3

from

- 3 csc (x) + 6 : - 3 (csc (x) - 2)

- 3 csc (x) + 6

Rewrite

6

as

2 \cdot 3

= - 3 csc (x) + 2 \cdot 3

Factor out common term

- 3

= - 3 (csc (x) - 2)

Factor out

tan (x)

from

tan (x) csc (x) - 2 tan (x) : tan (x) (csc (x) - 2)

tan (x) csc (x) - 2 tan (x)

Factor out common term

tan (x)

= tan (x) (csc (x) - 2)

= - 3 (csc (x) - 2) + tan (x) (csc (x) - 2)

Factor out common term

csc (x) - 2

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

(csc (x) - 2) (tan (x) - 3) = 0

Solving each part separately

csc (x) - 2 = 0 or tan (x) - 3 = 0

csc (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) - 2 = 0

Move

2

to the right side

csc (x) - 2 = 0

Add

2

to both sides

csc (x) - 2 + 2 = 0 + 2

Simplify

csc (x) = 2

csc (x) = 2

General solutions for

csc (x) = 2

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan (x) - 3 = 0 : x = arctan (3) + πn

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

tan (x) - 3 + 3 = 0 + 3

Simplify

tan (x) = 3

tan (x) = 3

Apply trig inverse properties

tan (x) = 3

General solutions for

tan (x) = 3

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

x = arctan (3) + πn

x = arctan (3) + πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arctan (3) + πn

Show solutions in decimal form

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 1.24904 \dots + πn

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