What is the general solution for sqrt(3csc(x))=2 ?

The general solution for sqrt(3csc(x))=2 is x=0.84806…+2pin,x=pi-0.84806…+2pin

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Popular >

sqrt(3csc(x))=2

Solution

3 csc (x) = 2

Solution

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Radians

x = 0.84806 \dots + 6.28318 \dots n, x = π - 0.84806 \dots + 6.28318 \dots n

Solution steps

3 csc (x) = 2

Square both sides:

3 csc (x) = 4

3 csc (x) = 2

(3 csc (x))^{2} = 2^{2}

Expand

(3 csc (x))^{2} : 3 csc (x)

(3 csc (x))^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((3 csc (x))^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (3 csc (x))^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3 csc (x)

Expand

2^{2} : 4

2^{2}

2^{2} = 4

= 4

3 csc (x) = 4

3 csc (x) = 4

Solve

3 csc (x) = 4 : csc (x) = \frac{4}{3}

3 csc (x) = 4

Divide both sides by

3

3 csc (x) = 4

Divide both sides by

3

\frac{3 csc ( x )}{3} = \frac{4}{3}

Simplify

csc (x) = \frac{4}{3}

csc (x) = \frac{4}{3}

csc (x) = \frac{4}{3}

Verify Solutions:

csc (x) = \frac{4}{3}

True

Check the solutions by plugging them into

3 csc (x) = 2

Remove the ones that don't agree with the equation.

Plug in

csc (x) = \frac{4}{3} :

True

3 (\frac{4}{3}) = 2

3 (\frac{4}{3}) = 2

3 (\frac{4}{3})

Remove parentheses:

(a) = a

= 3 \cdot \frac{4}{3}

Multiply

3 \cdot \frac{4}{3} : 4

3 \cdot \frac{4}{3}

Multiply fractions:

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

= \frac{4 \cdot 3}{3}

Cancel the common factor:

3

= 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

2 = 2

True

The solution is

csc (x) = \frac{4}{3}

Apply trig inverse properties

csc (x) = \frac{4}{3}

General solutions for

csc (x) = \frac{4}{3}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

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

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

Show solutions in decimal form

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Popular Examples

4csc^2(x)-25=0

4 csc^{2} (x) - 25 = 0

sin(θ)=-4/5 ,(3pi)/2 <θ<2pi

sin (θ) = - \frac{4}{5}, \frac{3 π}{2} < θ < 2 π

4cos(x)+sin^2(x)=3cos^2(x)+2

4 cos (x) + sin^{2} (x) = 3 cos^{2} (x) + 2

5tan(B)+sqrt(13)=0

5 tan (B) + 13 = 0

tan^2(x)=sqrt(3)tan(x)

tan^{2} (x) = 3 tan (x)

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3csc(x))=2 ?
The general solution for sqrt(3csc(x))=2 is x=0.84806…+2pin,x=pi-0.84806…+2pin