What is the general solution for 10=6.7+13.75tan(θ)-4.718(1+tan^2(θ)) ?

The general solution for 10=6.7+13.75tan(θ)-4.718(1+tan^2(θ)) is θ=0.67843…+pin,θ=1.12790…+pin

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10=6.7+13.75tan(θ)-4.718(1+tan^2(θ))

Solution

10 = 6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ))

Solution

θ = 0.67843 \dots + πn, θ = 1.12790 \dots + πn

Degrees

θ = 38.87154 \dots^{\circ} + 18 0^{\circ} n, θ = 64.62419 \dots^{\circ} + 18 0^{\circ} n

Solution steps

10 = 6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ))

Switch sides

6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ)) = 10

Solve by substitution

6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ)) = 10

Let:

tan (θ) = u

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10 : u = \frac{6875 - 9436701}{4718}, u = \frac{6875 + 9436701}{4718}

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

Multiply both sides by

1000

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

3

digits to the right of the decimal point, therefore multiply by

1000

6.7 \cdot 1000 + 13.75 u \cdot 1000 - 4.718 (1 + u^{2}) \cdot 1000 = 10 \cdot 1000

Refine

6700 + 13750 u - 4718 (1 + u^{2}) = 10000

6700 + 13750 u - 4718 (1 + u^{2}) = 10000

Expand

6700 + 13750 u - 4718 (1 + u^{2}) : - 4718 u^{2} + 13750 u + 1982

6700 + 13750 u - 4718 (1 + u^{2})

Expand

- 4718 (1 + u^{2}) : - 4718 - 4718 u^{2}

- 4718 (1 + u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = - 4718, b = 1, c = u^{2}

= - 4718 \cdot 1 + (- 4718) u^{2}

Apply minus-plus rules

+ (- a) = - a

= - 4718 \cdot 1 - 4718 u^{2}

Multiply the numbers:

4718 \cdot 1 = 4718

= - 4718 - 4718 u^{2}

= 6700 + 13750 u - 4718 - 4718 u^{2}

Simplify

6700 + 13750 u - 4718 - 4718 u^{2} : - 4718 u^{2} + 13750 u + 1982

6700 + 13750 u - 4718 - 4718 u^{2}

Group like terms

= - 4718 u^{2} + 13750 u + 6700 - 4718

Add/Subtract the numbers:

6700 - 4718 = 1982

= - 4718 u^{2} + 13750 u + 1982

= - 4718 u^{2} + 13750 u + 1982

- 4718 u^{2} + 13750 u + 1982 = 10000

Move

10000

to the left side

- 4718 u^{2} + 13750 u + 1982 = 10000

Subtract

10000

from both sides

- 4718 u^{2} + 13750 u + 1982 - 10000 = 10000 - 10000

Simplify

- 4718 u^{2} + 13750 u - 8018 = 0

- 4718 u^{2} + 13750 u - 8018 = 0

Solve with the quadratic formula

- 4718 u^{2} + 13750 u - 8018 = 0

Quadratic Equation Formula:

For

a = - 4718, b = 13750, c = - 8018

u_{1, 2} = \frac{- 13750 \pm 1375 0 ^{2} - 4 ( - 4718 ) ( - 8018 )}{2 ( - 4718 )}

u_{1, 2} = \frac{- 13750 \pm 1375 0 ^{2} - 4 ( - 4718 ) ( - 8018 )}{2 ( - 4718 )}

1375 0^{2} - 4 (- 4718) (- 8018) = 29436701

1375 0^{2} - 4 (- 4718) (- 8018)

Apply rule

- (- a) = a

= 1375 0^{2} - 4 \cdot 4718 \cdot 8018

Multiply the numbers:

4 \cdot 4718 \cdot 8018 = 151315696

= 1375 0^{2} - 151315696

1375 0^{2} = 189062500

= 189062500 - 151315696

Subtract the numbers:

189062500 - 151315696 = 37746804

= 37746804

Prime factorization of

37746804 : 2^{2} \cdot 3 \cdot 179 \cdot 17573

37746804

= 2^{2} \cdot 3 \cdot 179 \cdot 17573

Apply radical rule:

n ab = n a n b

= 2^{2} 3 \cdot 179 \cdot 17573

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 179 \cdot 17573

Refine

= 29436701

u_{1, 2} = \frac{- 13750 \pm 2 9436701}{2 ( - 4718 )}

Separate the solutions

u_{1} = \frac{- 13750 + 2 9436701}{2 ( - 4718 )}, u_{2} = \frac{- 13750 - 2 9436701}{2 ( - 4718 )}

u = \frac{- 13750 + 2 9436701}{2 ( - 4718 )} : \frac{6875 - 9436701}{4718}

\frac{- 13750 + 2 9436701}{2 ( - 4718 )}

Remove parentheses:

(- a) = - a

= \frac{- 13750 + 2 9436701}{- 2 \cdot 4718}

Multiply the numbers:

2 \cdot 4718 = 9436

= \frac{- 13750 + 2 9436701}{- 9436}

Apply the fraction rule:

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

- 13750 + 29436701 = - (13750 - 29436701)

= \frac{13750 - 2 9436701}{9436}

Factor

13750 - 29436701 : 2 (6875 - 9436701)

13750 - 29436701

Rewrite as

= 2 \cdot 6875 - 29436701

Factor out common term

2

= 2 (6875 - 9436701)

= \frac{2 ( 6875 - 9436701 )}{9436}

Cancel the common factor:

2

= \frac{6875 - 9436701}{4718}

u = \frac{- 13750 - 2 9436701}{2 ( - 4718 )} : \frac{6875 + 9436701}{4718}

\frac{- 13750 - 2 9436701}{2 ( - 4718 )}

Remove parentheses:

(- a) = - a

= \frac{- 13750 - 2 9436701}{- 2 \cdot 4718}

Multiply the numbers:

2 \cdot 4718 = 9436

= \frac{- 13750 - 2 9436701}{- 9436}

Apply the fraction rule:

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

- 13750 - 29436701 = - (13750 + 29436701)

= \frac{13750 + 2 9436701}{9436}

Factor

13750 + 29436701 : 2 (6875 + 9436701)

13750 + 29436701

Rewrite as

= 2 \cdot 6875 + 29436701

Factor out common term

2

= 2 (6875 + 9436701)

= \frac{2 ( 6875 + 9436701 )}{9436}

Cancel the common factor:

2

= \frac{6875 + 9436701}{4718}

The solutions to the quadratic equation are:

u = \frac{6875 - 9436701}{4718}, u = \frac{6875 + 9436701}{4718}

Substitute back

u = tan (θ)

tan (θ) = \frac{6875 - 9436701}{4718}, tan (θ) = \frac{6875 + 9436701}{4718}

tan (θ) = \frac{6875 - 9436701}{4718}, tan (θ) = \frac{6875 + 9436701}{4718}

tan (θ) = \frac{6875 - 9436701}{4718} : θ = arctan (\frac{6875 - 9436701}{4718}) + πn

tan (θ) = \frac{6875 - 9436701}{4718}

Apply trig inverse properties

tan (θ) = \frac{6875 - 9436701}{4718}

General solutions for

tan (θ) = \frac{6875 - 9436701}{4718}

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

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

tan (θ) = \frac{6875 + 9436701}{4718} : θ = arctan (\frac{6875 + 9436701}{4718}) + πn

tan (θ) = \frac{6875 + 9436701}{4718}

Apply trig inverse properties

tan (θ) = \frac{6875 + 9436701}{4718}

General solutions for

tan (θ) = \frac{6875 + 9436701}{4718}

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

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

Combine all the solutions

θ = arctan (\frac{6875 - 9436701}{4718}) + πn, θ = arctan (\frac{6875 + 9436701}{4718}) + πn

Show solutions in decimal form

θ = 0.67843 \dots + πn, θ = 1.12790 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 10=6.7+13.75tan(θ)-4.718(1+tan^2(θ)) ?
The general solution for 10=6.7+13.75tan(θ)-4.718(1+tan^2(θ)) is θ=0.67843…+pin,θ=1.12790…+pin