What is the general solution for 4sin(θ)+3csc(θ)=7 ?

The general solution for 4sin(θ)+3csc(θ)=7 is θ=0.84806…+2pin,θ=pi-0.84806…+2pin,θ= pi/2+2pin

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4sin(θ)+3csc(θ)=7

Solution

4 sin (θ) + 3 csc (θ) = 7

Solution

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

Degrees

θ = 48.59037 \dots^{\circ} + 36 0^{\circ} n, θ = 131.40962 \dots^{\circ} + 36 0^{\circ} n, θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (θ) + 3 csc (θ) = 7

Subtract

7

from both sides

4 sin (θ) + 3 csc (θ) - 7 = 0

Rewrite using trig identities

- 7 + 3 csc (θ) + 4 sin (θ)

Use the basic trigonometric identity:

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

= - 7 + 3 csc (θ) + 4 \cdot \frac{1}{csc ( θ )}

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

4 \cdot \frac{1}{csc ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 4}{csc ( θ )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{csc ( θ )}

= - 7 + 3 csc (θ) + \frac{4}{csc ( θ )}

- 7 + \frac{4}{csc ( θ )} + 3 csc (θ) = 0

Solve by substitution

- 7 + \frac{4}{csc ( θ )} + 3 csc (θ) = 0

Let:

csc (θ) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

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

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

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

Solve

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

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

Write in the standard form

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

3 u^{2} - 7 u + 4 = 0

Solve with the quadratic formula

3 u^{2} - 7 u + 4 = 0

Quadratic Equation Formula:

For

a = 3, b = - 7, c = 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

Subtract the numbers:

49 - 48 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

7 + 1 = 8

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

7 - 1 = 6

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = csc (θ)

csc (θ) = \frac{4}{3}, csc (θ) = 1

csc (θ) = \frac{4}{3}, csc (θ) = 1

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

csc (θ) = \frac{4}{3}

Apply trig inverse properties

csc (θ) = \frac{4}{3}

General solutions for

csc (θ) = \frac{4}{3}

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

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

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

csc (θ) = 1 : θ = \frac{π}{2} + 2 πn

csc (θ) = 1

General solutions for

csc (θ) = 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

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Combine all the solutions

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

Show solutions in decimal form

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

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin(θ)+3csc(θ)=7 ?
The general solution for 4sin(θ)+3csc(θ)=7 is θ=0.84806…+2pin,θ=pi-0.84806…+2pin,θ= pi/2+2pin