What is the general solution for -4sec^2(x)+19=-4tan(x) ?

The general solution for -4sec^2(x)+19=-4tan(x) is x=-0.98279…+pin,x=1.19028…+pin

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-4sec^2(x)+19=-4tan(x)

Solution

- 4 sec^{2} (x) + 19 = - 4 tan (x)

Solution

x = - 0.98279 \dots + πn, x = 1.19028 \dots + πn

Degrees

x = - 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = 68.19859 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 4 sec^{2} (x) + 19 = - 4 tan (x)

Subtract

- 4 tan (x)

from both sides

- 4 sec^{2} (x) + 19 + 4 tan (x) = 0

Rewrite using trig identities

19 - 4 sec^{2} (x) + 4 tan (x)

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

= 19 - 4 (tan^{2} (x) + 1) + 4 tan (x)

Simplify

19 - 4 (tan^{2} (x) + 1) + 4 tan (x) : 4 tan (x) - 4 tan^{2} (x) + 15

19 - 4 (tan^{2} (x) + 1) + 4 tan (x)

Expand

- 4 (tan^{2} (x) + 1) : - 4 tan^{2} (x) - 4

- 4 (tan^{2} (x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - 4, b = tan^{2} (x), c = 1

= - 4 tan^{2} (x) + (- 4) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 4 tan^{2} (x) - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= - 4 tan^{2} (x) - 4

= 19 - 4 tan^{2} (x) - 4 + 4 tan (x)

Simplify

19 - 4 tan^{2} (x) - 4 + 4 tan (x) : 4 tan (x) - 4 tan^{2} (x) + 15

19 - 4 tan^{2} (x) - 4 + 4 tan (x)

Group like terms

= - 4 tan^{2} (x) + 4 tan (x) + 19 - 4

Add/Subtract the numbers:

19 - 4 = 15

= 4 tan (x) - 4 tan^{2} (x) + 15

= 4 tan (x) - 4 tan^{2} (x) + 15

= 4 tan (x) - 4 tan^{2} (x) + 15

15 + 4 tan (x) - 4 tan^{2} (x) = 0

Solve by substitution

15 + 4 tan (x) - 4 tan^{2} (x) = 0

Let:

tan (x) = u

15 + 4 u - 4 u^{2} = 0

15 + 4 u - 4 u^{2} = 0 : u = - \frac{3}{2}, u = \frac{5}{2}

15 + 4 u - 4 u^{2} = 0

Write in the standard form

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

- 4 u^{2} + 4 u + 15 = 0

Solve with the quadratic formula

- 4 u^{2} + 4 u + 15 = 0

Quadratic Equation Formula:

For

a = - 4, b = 4, c = 15

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 4 ) \cdot 15}{2 ( - 4 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 4 ) \cdot 15}{2 ( - 4 )}

4^{2} - 4 (- 4) \cdot 15 = 16

4^{2} - 4 (- 4) \cdot 15

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 4 \cdot 15

Multiply the numbers:

4 \cdot 4 \cdot 15 = 240

= 4^{2} + 240

4^{2} = 16

= 16 + 240

Add the numbers:

16 + 240 = 256

= 256

Factor the number:

256 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

n a^{n} = a

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- 4 \pm 16}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 4 + 16}{2 ( - 4 )}, u_{2} = \frac{- 4 - 16}{2 ( - 4 )}

u = \frac{- 4 + 16}{2 ( - 4 )} : - \frac{3}{2}

\frac{- 4 + 16}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 16}{- 2 \cdot 4}

Add/Subtract the numbers:

- 4 + 16 = 12

= \frac{12}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{12}{- 8}

Apply the fraction rule:

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

= - \frac{12}{8}

Cancel the common factor:

4

= - \frac{3}{2}

u = \frac{- 4 - 16}{2 ( - 4 )} : \frac{5}{2}

\frac{- 4 - 16}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 16}{- 2 \cdot 4}

Subtract the numbers:

- 4 - 16 = - 20

= \frac{- 20}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 20}{- 8}

Apply the fraction rule:

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

= \frac{20}{8}

Cancel the common factor:

4

= \frac{5}{2}

The solutions to the quadratic equation are:

u = - \frac{3}{2}, u = \frac{5}{2}

Substitute back

u = tan (x)

tan (x) = - \frac{3}{2}, tan (x) = \frac{5}{2}

tan (x) = - \frac{3}{2}, tan (x) = \frac{5}{2}

tan (x) = - \frac{3}{2} : x = arctan (- \frac{3}{2}) + πn

tan (x) = - \frac{3}{2}

Apply trig inverse properties

tan (x) = - \frac{3}{2}

General solutions for

tan (x) = - \frac{3}{2}

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

tan (x) = \frac{5}{2} : x = arctan (\frac{5}{2}) + πn

tan (x) = \frac{5}{2}

Apply trig inverse properties

tan (x) = \frac{5}{2}

General solutions for

tan (x) = \frac{5}{2}

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

x = arctan (\frac{5}{2}) + πn

x = arctan (\frac{5}{2}) + πn

Combine all the solutions

x = arctan (- \frac{3}{2}) + πn, x = arctan (\frac{5}{2}) + πn

Show solutions in decimal form

x = - 0.98279 \dots + πn, x = 1.19028 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -4sec^2(x)+19=-4tan(x) ?
The general solution for -4sec^2(x)+19=-4tan(x) is x=-0.98279…+pin,x=1.19028…+pin