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Popular Trigonometry >

csc(x)=4sin(x)

Solution

csc (x) = 4 sin (x)

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

csc (x) = 4 sin (x)

Subtract

4 sin (x)

from both sides

csc (x) - 4 sin (x) = 0

Rewrite using trig identities

csc (x) - 4 sin (x)

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= csc (x) - 4 \cdot \frac{1}{csc ( x )}

4 \cdot \frac{1}{csc ( x )} = \frac{4}{csc ( x )}

4 \cdot \frac{1}{csc ( x )}

Multiply fractions:

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

= \frac{1 \cdot 4}{csc ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{csc ( x )}

= csc (x) - \frac{4}{csc ( x )}

csc (x) - \frac{4}{csc ( x )} = 0

Solve by substitution

csc (x) - \frac{4}{csc ( x )} = 0

Let:

csc (x) = u

u - \frac{4}{u} = 0

u - \frac{4}{u} = 0 : u = 2, u = - 2

u - \frac{4}{u} = 0

Multiply both sides by

u

u - \frac{4}{u} = 0

Multiply both sides by

u

uu - \frac{4}{u} u = 0 \cdot u

Simplify

uu - \frac{4}{u} u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

- \frac{4}{u} u : - 4

- \frac{4}{u} u

Multiply fractions:

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

= - \frac{4 u}{u}

Cancel the common factor:

u

= - 4

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 4 = 0

u^{2} - 4 = 0

u^{2} - 4 = 0

Solve

u^{2} - 4 = 0 : u = 2, u = - 2

u^{2} - 4 = 0

Move

4

to the right side

u^{2} - 4 = 0

Add

4

to both sides

u^{2} - 4 + 4 = 0 + 4

Simplify

u^{2} = 4

u^{2} = 4

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

u = 2, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - \frac{4}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 2

Substitute back

u = csc (x)

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

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

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

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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