What is the general solution for 4cot(x)=sqrt(3)csc(x) ?

The general solution for 4cot(x)=sqrt(3)csc(x) is x=1.12296…+2pin,x=2pi-1.12296…+2pin

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4cot(x)=sqrt(3)csc(x)

Solution

4 cot (x) = 3 csc (x)

Solution

x = 1.12296 \dots + 2 πn, x = 2 π - 1.12296 \dots + 2 πn

Degrees

x = 64.34109 \dots^{\circ} + 36 0^{\circ} n, x = 295.65890 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 cot (x) = 3 csc (x)

Subtract

3 csc (x)

from both sides

4 cot (x) - 3 csc (x) = 0

Express with sin, cos

4 cot (x) - csc (x) 3

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 4 \cdot \frac{cos ( x )}{sin ( x )} - csc (x) 3

Use the basic trigonometric identity:

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

= 4 \cdot \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} 3

Simplify

4 \cdot \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} 3 : \frac{4 cos ( x ) - 3}{sin ( x )}

4 \cdot \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} 3

4 \cdot \frac{cos ( x )}{sin ( x )} = \frac{4 cos ( x )}{sin ( x )}

4 \cdot \frac{cos ( x )}{sin ( x )}

Multiply fractions:

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

= \frac{cos ( x ) \cdot 4}{sin ( x )}

\frac{1}{sin ( x )} 3 = \frac{3}{sin ( x )}

\frac{1}{sin ( x )} 3

Multiply fractions:

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

= \frac{1 \cdot 3}{sin ( x )}

Multiply:

1 \cdot 3 = 3

= \frac{3}{sin ( x )}

= \frac{4 cos ( x )}{sin ( x )} - \frac{3}{sin ( x )}

Apply rule

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

= \frac{4 cos ( x ) - 3}{sin ( x )}

= \frac{4 cos ( x ) - 3}{sin ( x )}

\frac{- 3 + 4 cos ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 3 + 4 cos (x) = 0

Move

3

to the right side

- 3 + 4 cos (x) = 0

Add

3

to both sides

- 3 + 4 cos (x) + 3 = 0 + 3

Simplify

4 cos (x) = 3

4 cos (x) = 3

Divide both sides by

4

4 cos (x) = 3

Divide both sides by

4

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Show solutions in decimal form

x = 1.12296 \dots + 2 πn, x = 2 π - 1.12296 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4cot(x)=sqrt(3)csc(x) ?
The general solution for 4cot(x)=sqrt(3)csc(x) is x=1.12296…+2pin,x=2pi-1.12296…+2pin