What is the general solution for sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8) ?

The general solution for sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8) is θ=0.96256…+pin

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Popular Trigonometry >

sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8)

Solution

5 tan (θ) - 9 = - 6 tan (θ) + 8

Solution

θ = 0.96256 \dots + πn

Degrees

θ = 55.15072 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 tan (θ) - 9 = - 6 tan (θ) + 8

Solve by substitution

5 tan (θ) - 9 = - 6 tan (θ) + 8

Let:

tan (θ) = u

5 u - 9 = - 6 u + 8

5 u - 9 = - 6 u + 8 : u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

5 u - 9 = - 6 u + 8

Simplify

- 6 u + 8 : - 6 u + 22

- 6 u + 8

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= - 6 u + 22

5 u - 9 = - 6 u + 22

Move

9

to the right side

5 u - 9 = - 6 u + 22

Add

9

to both sides

5 u - 9 + 9 = - 6 u + 22 + 9

Simplify

5 u = - 6 u + 22 + 9

5 u = - 6 u + 22 + 9

Move

6 u

to the left side

5 u = - 6 u + 22 + 9

Add

6 u

to both sides

5 u + 6 u = - 6 u + 22 + 9 + 6 u

Simplify

5 u + 6 u = 22 + 9

5 u + 6 u = 22 + 9

Factor

5 u + 6 u : (5 + 6) u

5 u + 6 u

Factor out common term

u

= u (5 + 6)

(5 + 6) u = 22 + 9

Divide both sides by

5 + 6

(5 + 6) u = 22 + 9

Divide both sides by

5 + 6

\frac{( 5 + 6 ) u}{5 + 6} = \frac{2 2}{5 + 6} + \frac{9}{5 + 6}

Simplify

\frac{( 5 + 6 ) u}{5 + 6} = \frac{2 2}{5 + 6} + \frac{9}{5 + 6}

Simplify

\frac{( 5 + 6 ) u}{5 + 6} : u

\frac{( 5 + 6 ) u}{5 + 6}

Cancel the common factor:

5 + 6

= u

Simplify

\frac{2 2}{5 + 6} + \frac{9}{5 + 6} : - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

\frac{2 2}{5 + 6} + \frac{9}{5 + 6}

Apply rule

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

= \frac{2 2 + 9}{5 + 6}

Multiply by the conjugate

\frac{5 - 6}{5 - 6}

= \frac{( 2 2 + 9 ) ( 5 - 6 )}{( 5 + 6 ) ( 5 - 6 )}

(5 + 6) (5 - 6) = - 31

(5 + 6) (5 - 6)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 5, b = 6

= (5)^{2} - 6^{2}

Simplify

(5)^{2} - 6^{2} : - 31

(5)^{2} - 6^{2}

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 5^{\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

= 5

6^{2} = 36

6^{2}

6^{2} = 36

= 36

= 5 - 36

Subtract the numbers:

5 - 36 = - 31

= - 31

= - 31

= \frac{( 2 2 + 9 ) ( 5 - 6 )}{- 31}

Apply the fraction rule:

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

= - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

Substitute back

u = tan (θ)

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31} : θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

Apply trig inverse properties

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

General solutions for

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (x) = a \Rightarrow x = arctan (a) + πn

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

Combine all the solutions

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

Show solutions in decimal form

θ = 0.96256 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8) ?
The general solution for sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8) is θ=0.96256…+pin