What is the general solution for (9tan(θ))/(10-tan^2(θ))=1 ?

The general solution for (9tan(θ))/(10-tan^2(θ))=1 is θ=-1.47112…+pin,θ= pi/4+pin

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(9tan(θ))/(10-tan^2(θ))=1

Solution

\frac{9 tan ( θ )}{10 - tan ^{2} ( θ )} = 1

Solution

θ = - 1.47112 \dots + πn, θ = \frac{π}{4} + πn

Degrees

θ = - 84.28940 \dots^{\circ} + 18 0^{\circ} n, θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{9 tan ( θ )}{10 - tan ^{2} ( θ )} = 1

Solve by substitution

\frac{9 tan ( θ )}{10 - tan ^{2} ( θ )} = 1

Let:

tan (θ) = u

\frac{9 u}{10 - u ^{2}} = 1

\frac{9 u}{10 - u ^{2}} = 1 : u = - 10, u = 1

\frac{9 u}{10 - u ^{2}} = 1

Multiply both sides by

10 - u^{2}

\frac{9 u}{10 - u ^{2}} = 1

Multiply both sides by

10 - u^{2}

\frac{9 u}{10 - u ^{2}} (10 - u^{2}) = 1 \cdot (10 - u^{2})

Simplify

\frac{9 u}{10 - u ^{2}} (10 - u^{2}) = 1 \cdot (10 - u^{2})

Simplify

\frac{9 u}{10 - u ^{2}} (10 - u^{2}) : 9 u

\frac{9 u}{10 - u ^{2}} (10 - u^{2})

Multiply fractions:

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

= \frac{9 u ( 10 - u ^{2} )}{10 - u ^{2}}

Cancel the common factor:

10 - u^{2}

= 9 u

Simplify

1 \cdot (10 - u^{2}) : 10 - u^{2}

1 \cdot (10 - u^{2})

Multiply:

1 \cdot (10 - u^{2}) = (10 - u^{2})

= (10 - u^{2})

Remove parentheses:

(a) = a

= 10 - u^{2}

9 u = 10 - u^{2}

9 u = 10 - u^{2}

9 u = 10 - u^{2}

Solve

9 u = 10 - u^{2} : u = - 10, u = 1

9 u = 10 - u^{2}

Switch sides

10 - u^{2} = 9 u

Move

9 u

to the left side

10 - u^{2} = 9 u

Subtract

9 u

from both sides

10 - u^{2} - 9 u = 9 u - 9 u

Simplify

10 - u^{2} - 9 u = 0

10 - u^{2} - 9 u = 0

Write in the standard form

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

- u^{2} - 9 u + 10 = 0

Solve with the quadratic formula

- u^{2} - 9 u + 10 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 9, c = 10

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 1 ) \cdot 10}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 1 ) \cdot 10}{2 ( - 1 )}

(- 9)^{2} - 4 (- 1) \cdot 10 = 11

(- 9)^{2} - 4 (- 1) \cdot 10

Apply rule

- (- a) = a

= (- 9)^{2} + 4 \cdot 1 \cdot 10

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 1 \cdot 10

Multiply the numbers:

4 \cdot 1 \cdot 10 = 40

= 9^{2} + 40

9^{2} = 81

= 81 + 40

Add the numbers:

81 + 40 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 9 ) \pm 11}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 11}{2 ( - 1 )}, u_{2} = \frac{- ( - 9 ) - 11}{2 ( - 1 )}

u = \frac{- ( - 9 ) + 11}{2 ( - 1 )} : - 10

\frac{- ( - 9 ) + 11}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{9 + 11}{- 2 \cdot 1}

Add the numbers:

9 + 11 = 20

= \frac{20}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{20}{- 2}

Apply the fraction rule:

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

= - \frac{20}{2}

Divide the numbers:

\frac{20}{2} = 10

= - 10

u = \frac{- ( - 9 ) - 11}{2 ( - 1 )} : 1

\frac{- ( - 9 ) - 11}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{9 - 11}{- 2 \cdot 1}

Subtract the numbers:

9 - 11 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

Apply the fraction rule:

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

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - 10, u = 1

u = - 10, u = 1

Verify Solutions

Find undefined (singularity) points:

u = 10, u = - 10

Take the denominator(s) of

\frac{9 u}{10 - u ^{2}}

and compare to zero

Solve

10 - u^{2} = 0 : u = 10, u = - 10

10 - u^{2} = 0

Move

10

to the right side

10 - u^{2} = 0

Subtract

10

from both sides

10 - u^{2} - 10 = 0 - 10

Simplify

- u^{2} = - 10

- u^{2} = - 10

Divide both sides by

- 1

- u^{2} = - 10

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 10}{- 1}

Simplify

u^{2} = 10

u^{2} = 10

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 10, u = - 10

The following points are undefined

u = 10, u = - 10

Combine undefined points with solutions:

u = - 10, u = 1

Substitute back

u = tan (θ)

tan (θ) = - 10, tan (θ) = 1

tan (θ) = - 10, tan (θ) = 1

tan (θ) = - 10 : θ = arctan (- 10) + πn

tan (θ) = - 10

Apply trig inverse properties

tan (θ) = - 10

General solutions for

tan (θ) = - 10

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

θ = arctan (- 10) + πn

θ = arctan (- 10) + πn

tan (θ) = 1 : θ = \frac{π}{4} + πn

tan (θ) = 1

General solutions for

tan (θ) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

Combine all the solutions

θ = arctan (- 10) + πn, θ = \frac{π}{4} + πn

Show solutions in decimal form

θ = - 1.47112 \dots + πn, θ = \frac{π}{4} + πn

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6 sin (2 x) = 3 cos (3 0^{\circ})

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Frequently Asked Questions (FAQ)

What is the general solution for (9tan(θ))/(10-tan^2(θ))=1 ?
The general solution for (9tan(θ))/(10-tan^2(θ))=1 is θ=-1.47112…+pin,θ= pi/4+pin