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Popular Trigonometry >

csc(7x+3)=sqrt(2)

Solution

csc (7 x + 3) = 2

Solution

x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}, x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

+1

Degrees

x = - 18.12676 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = - 5.26961 \dots^{\circ} + 51.42857 \dots^{\circ} n

Solution steps

csc (7 x + 3) = 2

General solutions for

csc (7 x + 3) = 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

7 x + 3 = \frac{π}{4} + 2 πn, 7 x + 3 = \frac{3 π}{4} + 2 πn

7 x + 3 = \frac{π}{4} + 2 πn, 7 x + 3 = \frac{3 π}{4} + 2 πn

Solve

7 x + 3 = \frac{π}{4} + 2 πn : x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

7 x + 3 = \frac{π}{4} + 2 πn

Move

3

to the right side

7 x + 3 = \frac{π}{4} + 2 πn

Subtract

3

from both sides

7 x + 3 - 3 = \frac{π}{4} + 2 πn - 3

Simplify

7 x = \frac{π}{4} + 2 πn - 3

7 x = \frac{π}{4} + 2 πn - 3

Divide both sides by

7

7 x = \frac{π}{4} + 2 πn - 3

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7} : - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

\frac{\frac{π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Group like terms

= - \frac{3}{7} + \frac{2 πn}{7} + \frac{\frac{π}{4}}{7}

\frac{\frac{π}{4}}{7} = \frac{π}{28}

\frac{\frac{π}{4}}{7}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{4 \cdot 7}

Multiply the numbers:

4 \cdot 7 = 28

= \frac{π}{28}

= - \frac{3}{7} + \frac{2 πn}{7} + \frac{π}{28}

Group like terms

= - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}

Solve

7 x + 3 = \frac{3 π}{4} + 2 πn : x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

7 x + 3 = \frac{3 π}{4} + 2 πn

Move

3

to the right side

7 x + 3 = \frac{3 π}{4} + 2 πn

Subtract

3

from both sides

7 x + 3 - 3 = \frac{3 π}{4} + 2 πn - 3

Simplify

7 x = \frac{3 π}{4} + 2 πn - 3

7 x = \frac{3 π}{4} + 2 πn - 3

Divide both sides by

7

7 x = \frac{3 π}{4} + 2 πn - 3

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{3 π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{3 π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7} : - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

\frac{\frac{3 π}{4}}{7} + \frac{2 πn}{7} - \frac{3}{7}

Group like terms

= - \frac{3}{7} + \frac{2 πn}{7} + \frac{\frac{3 π}{4}}{7}

\frac{\frac{3 π}{4}}{7} = \frac{3 π}{28}

\frac{\frac{3 π}{4}}{7}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π}{4 \cdot 7}

Multiply the numbers:

4 \cdot 7 = 28

= \frac{3 π}{28}

= - \frac{3}{7} + \frac{2 πn}{7} + \frac{3 π}{28}

Group like terms

= - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

x = - \frac{3}{7} + \frac{π}{28} + \frac{2 πn}{7}, x = - \frac{3}{7} + \frac{3 π}{28} + \frac{2 πn}{7}

Graph

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