What is the general solution for 4sin(x)+1=3csc(x) ?

The general solution for 4sin(x)+1=3csc(x) is x=(3pi)/2+2pin,x=0.84806…+2pin,x=pi-0.84806…+2pin

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4sin(x)+1=3csc(x)

Solution

4 sin (x) + 1 = 3 csc (x)

Solution

x = \frac{3 π}{2} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (x) + 1 = 3 csc (x)

Subtract

3 csc (x)

from both sides

4 sin (x) + 1 - 3 csc (x) = 0

Rewrite using trig identities

1 - 3 csc (x) + 4 sin (x)

Use the basic trigonometric identity:

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

= 1 - 3 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 )}

= 1 - 3 csc (x) + \frac{4}{csc ( x )}

1 + \frac{4}{csc ( x )} - 3 csc (x) = 0

Solve by substitution

1 + \frac{4}{csc ( x )} - 3 csc (x) = 0

Let:

csc (x) = u

1 + \frac{4}{u} - 3 u = 0

1 + \frac{4}{u} - 3 u = 0 : u = - 1, u = \frac{4}{3}

1 + \frac{4}{u} - 3 u = 0

Multiply both sides by

u

1 + \frac{4}{u} - 3 u = 0

Multiply both sides by

u

1 \cdot u + \frac{4}{u} u - 3 uu = 0 \cdot u

Simplify

1 \cdot u + \frac{4}{u} u - 3 uu = 0 \cdot u

Simplify

1 \cdot u : u

1 \cdot u

Multiply:

1 \cdot u = u

= u

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

- 3 uu : - 3 u^{2}

- 3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u + 4 - 3 u^{2} = 0

u + 4 - 3 u^{2} = 0

u + 4 - 3 u^{2} = 0

Solve

u + 4 - 3 u^{2} = 0 : u = - 1, u = \frac{4}{3}

u + 4 - 3 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 3, b = 1, c = 4

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

1^{2} - 4 (- 3) \cdot 4 = 7

1^{2} - 4 (- 3) \cdot 4

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 3) \cdot 4

Apply rule

- (- a) = a

= 1 + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 1 \pm 7}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 1 + 7}{2 ( - 3 )}, u_{2} = \frac{- 1 - 7}{2 ( - 3 )}

u = \frac{- 1 + 7}{2 ( - 3 )} : - 1

\frac{- 1 + 7}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 7}{- 2 \cdot 3}

Add/Subtract the numbers:

- 1 + 7 = 6

= \frac{6}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- 1 - 7}{2 ( - 3 )} : \frac{4}{3}

\frac{- 1 - 7}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 7}{- 2 \cdot 3}

Subtract the numbers:

- 1 - 7 = - 8

= \frac{- 8}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 8}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

The solutions to the quadratic equation are:

u = - 1, u = \frac{4}{3}

u = - 1, u = \frac{4}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 + \frac{4}{u} - 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1, u = \frac{4}{3}

Substitute back

u = csc (x)

csc (x) = - 1, csc (x) = \frac{4}{3}

csc (x) = - 1, csc (x) = \frac{4}{3}

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

General solutions for

csc (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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

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

Combine all the solutions

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

Show solutions in decimal form

x = \frac{3 π}{2} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Popular Examples

cos^2(x)-3sin(x)=0

cos^{2} (x) - 3 sin (x) = 0

csc(t)= 1/(-(sqrt(2))/2)

csc (t) = \frac{1}{- \frac{2}{2}}

cos^2(x)=1-sin(x),0<= x<2pi

cos^{2} (x) = 1 - sin (x), 0 \leq x < 2 π

cos(2x)-3sin(x)=2,0<= x<= 360

cos (2 x) - 3 sin (x) = 2, 0^{\circ} \leq x \leq 36 0^{\circ}

2cos^2(x)+3cos(x)=4cos(x)+1

2 cos^{2} (x) + 3 cos (x) = 4 cos (x) + 1

Frequently Asked Questions (FAQ)

What is the general solution for 4sin(x)+1=3csc(x) ?
The general solution for 4sin(x)+1=3csc(x) is x=(3pi)/2+2pin,x=0.84806…+2pin,x=pi-0.84806…+2pin