What is the general solution for 8.59tan^2(θ)-24tan(θ)+15.29=0 ?

The general solution for 8.59tan^2(θ)-24tan(θ)+15.29=0 is θ=1.06631…+pin,θ=0.77671…+pin

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8.59tan^2(θ)-24tan(θ)+15.29=0

Solution

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

Solution

θ = 1.06631 \dots + πn, θ = 0.77671 \dots + πn

Degrees

θ = 61.09560 \dots^{\circ} + 18 0^{\circ} n, θ = 44.50236 \dots^{\circ} + 18 0^{\circ} n

Solution steps

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

Solve by substitution

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

Let:

tan (θ) = u

8.59 u^{2} - 24 u + 15.29 = 0

8.59 u^{2} - 24 u + 15.29 = 0 : u = \frac{1200 + 126589}{859}, u = \frac{1200 - 126589}{859}

8.59 u^{2} - 24 u + 15.29 = 0

Multiply both sides by

100

8.59 u^{2} - 24 u + 15.29 = 0

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

2

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

100

8.59 u^{2} \cdot 100 - 24 u \cdot 100 + 15.29 \cdot 100 = 0 \cdot 100

Refine

859 u^{2} - 2400 u + 1529 = 0

859 u^{2} - 2400 u + 1529 = 0

Solve with the quadratic formula

859 u^{2} - 2400 u + 1529 = 0

Quadratic Equation Formula:

For

a = 859, b = - 2400, c = 1529

u_{1, 2} = \frac{- ( - 2400 ) \pm ( - 2400 ) ^{2} - 4 \cdot 859 \cdot 1529}{2 \cdot 859}

u_{1, 2} = \frac{- ( - 2400 ) \pm ( - 2400 ) ^{2} - 4 \cdot 859 \cdot 1529}{2 \cdot 859}

(- 2400)^{2} - 4 \cdot 859 \cdot 1529 = 2126589

(- 2400)^{2} - 4 \cdot 859 \cdot 1529

Apply exponent rule:

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

n

is even

(- 2400)^{2} = 240 0^{2}

= 240 0^{2} - 4 \cdot 859 \cdot 1529

Multiply the numbers:

4 \cdot 859 \cdot 1529 = 5253644

= 240 0^{2} - 5253644

240 0^{2} = 5760000

= 5760000 - 5253644

Subtract the numbers:

5760000 - 5253644 = 506356

= 506356

Prime factorization of

506356 : 2^{2} \cdot 277 \cdot 457

506356

= 2^{2} \cdot 277 \cdot 457

Apply radical rule:

= 2^{2} 277 \cdot 457

Apply radical rule:

2^{2} = 2

= 2 277 \cdot 457

Refine

= 2126589

u_{1, 2} = \frac{- ( - 2400 ) \pm 2 126589}{2 \cdot 859}

Separate the solutions

u_{1} = \frac{- ( - 2400 ) + 2 126589}{2 \cdot 859}, u_{2} = \frac{- ( - 2400 ) - 2 126589}{2 \cdot 859}

u = \frac{- ( - 2400 ) + 2 126589}{2 \cdot 859} : \frac{1200 + 126589}{859}

\frac{- ( - 2400 ) + 2 126589}{2 \cdot 859}

Apply rule

- (- a) = a

= \frac{2400 + 2 126589}{2 \cdot 859}

Multiply the numbers:

2 \cdot 859 = 1718

= \frac{2400 + 2 126589}{1718}

Factor

2400 + 2126589 : 2 (1200 + 126589)

2400 + 2126589

Rewrite as

= 2 \cdot 1200 + 2126589

Factor out common term

2

= 2 (1200 + 126589)

= \frac{2 ( 1200 + 126589 )}{1718}

Cancel the common factor:

2

= \frac{1200 + 126589}{859}

u = \frac{- ( - 2400 ) - 2 126589}{2 \cdot 859} : \frac{1200 - 126589}{859}

\frac{- ( - 2400 ) - 2 126589}{2 \cdot 859}

Apply rule

- (- a) = a

= \frac{2400 - 2 126589}{2 \cdot 859}

Multiply the numbers:

2 \cdot 859 = 1718

= \frac{2400 - 2 126589}{1718}

Factor

2400 - 2126589 : 2 (1200 - 126589)

2400 - 2126589

Rewrite as

= 2 \cdot 1200 - 2126589

Factor out common term

2

= 2 (1200 - 126589)

= \frac{2 ( 1200 - 126589 )}{1718}

Cancel the common factor:

2

= \frac{1200 - 126589}{859}

The solutions to the quadratic equation are:

u = \frac{1200 + 126589}{859}, u = \frac{1200 - 126589}{859}

Substitute back

u = tan (θ)

tan (θ) = \frac{1200 + 126589}{859}, tan (θ) = \frac{1200 - 126589}{859}

tan (θ) = \frac{1200 + 126589}{859}, tan (θ) = \frac{1200 - 126589}{859}

tan (θ) = \frac{1200 + 126589}{859} : θ = arctan (\frac{1200 + 126589}{859}) + πn

tan (θ) = \frac{1200 + 126589}{859}

Apply trig inverse properties

tan (θ) = \frac{1200 + 126589}{859}

General solutions for

tan (θ) = \frac{1200 + 126589}{859}

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

θ = arctan (\frac{1200 + 126589}{859}) + πn

θ = arctan (\frac{1200 + 126589}{859}) + πn

tan (θ) = \frac{1200 - 126589}{859} : θ = arctan (\frac{1200 - 126589}{859}) + πn

tan (θ) = \frac{1200 - 126589}{859}

Apply trig inverse properties

tan (θ) = \frac{1200 - 126589}{859}

General solutions for

tan (θ) = \frac{1200 - 126589}{859}

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

θ = arctan (\frac{1200 - 126589}{859}) + πn

θ = arctan (\frac{1200 - 126589}{859}) + πn

Combine all the solutions

θ = arctan (\frac{1200 + 126589}{859}) + πn, θ = arctan (\frac{1200 - 126589}{859}) + πn

Show solutions in decimal form

θ = 1.06631 \dots + πn, θ = 0.77671 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 8.59tan^2(θ)-24tan(θ)+15.29=0 ?
The general solution for 8.59tan^2(θ)-24tan(θ)+15.29=0 is θ=1.06631…+pin,θ=0.77671…+pin