What is the general solution for 18sec^2(x)-3tan(x)=21 ?

The general solution for 18sec^2(x)-3tan(x)=21 is x=0.46364…+pin,x=-0.32175…+pin

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Popular >

18sec^2(x)-3tan(x)=21

Solution

18 sec^{2} (x) - 3 tan (x) = 21

Solution

x = 0.46364 \dots + πn, x = - 0.32175 \dots + πn

Degrees

x = 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 18.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

18 sec^{2} (x) - 3 tan (x) = 21

Subtract

21

from both sides

18 sec^{2} (x) - 3 tan (x) - 21 = 0

Rewrite using trig identities

- 21 + 18 sec^{2} (x) - 3 tan (x)

Use the Pythagorean identity:

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

= - 21 + 18 (tan^{2} (x) + 1) - 3 tan (x)

Simplify

- 21 + 18 (tan^{2} (x) + 1) - 3 tan (x) : 18 tan^{2} (x) - 3 tan (x) - 3

- 21 + 18 (tan^{2} (x) + 1) - 3 tan (x)

Expand

18 (tan^{2} (x) + 1) : 18 tan^{2} (x) + 18

18 (tan^{2} (x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 18, b = tan^{2} (x), c = 1

= 18 tan^{2} (x) + 18 \cdot 1

Multiply the numbers:

18 \cdot 1 = 18

= 18 tan^{2} (x) + 18

= - 21 + 18 tan^{2} (x) + 18 - 3 tan (x)

Simplify

- 21 + 18 tan^{2} (x) + 18 - 3 tan (x) : 18 tan^{2} (x) - 3 tan (x) - 3

- 21 + 18 tan^{2} (x) + 18 - 3 tan (x)

Group like terms

= 18 tan^{2} (x) - 3 tan (x) - 21 + 18

Add/Subtract the numbers:

- 21 + 18 = - 3

= 18 tan^{2} (x) - 3 tan (x) - 3

= 18 tan^{2} (x) - 3 tan (x) - 3

= 18 tan^{2} (x) - 3 tan (x) - 3

- 3 + 18 tan^{2} (x) - 3 tan (x) = 0

Solve by substitution

- 3 + 18 tan^{2} (x) - 3 tan (x) = 0

Let:

tan (x) = u

- 3 + 18 u^{2} - 3 u = 0

- 3 + 18 u^{2} - 3 u = 0 : u = \frac{1}{2}, u = - \frac{1}{3}

- 3 + 18 u^{2} - 3 u = 0

Write in the standard form

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

18 u^{2} - 3 u - 3 = 0

Solve with the quadratic formula

18 u^{2} - 3 u - 3 = 0

Quadratic Equation Formula:

For

a = 18, b = - 3, c = - 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 18 ( - 3 )}{2 \cdot 18}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 18 ( - 3 )}{2 \cdot 18}

(- 3)^{2} - 4 \cdot 18 (- 3) = 15

(- 3)^{2} - 4 \cdot 18 (- 3)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 18 \cdot 3

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 18 \cdot 3

Multiply the numbers:

4 \cdot 18 \cdot 3 = 216

= 3^{2} + 216

3^{2} = 9

= 9 + 216

Add the numbers:

9 + 216 = 225

= 225

Factor the number:

225 = 1 5^{2}

= 1 5^{2}

Apply radical rule:

1 5^{2} = 15

= 15

u_{1, 2} = \frac{- ( - 3 ) \pm 15}{2 \cdot 18}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 15}{2 \cdot 18}, u_{2} = \frac{- ( - 3 ) - 15}{2 \cdot 18}

u = \frac{- ( - 3 ) + 15}{2 \cdot 18} : \frac{1}{2}

\frac{- ( - 3 ) + 15}{2 \cdot 18}

Apply rule

- (- a) = a

= \frac{3 + 15}{2 \cdot 18}

Add the numbers:

3 + 15 = 18

= \frac{18}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{18}{36}

Cancel the common factor:

18

= \frac{1}{2}

u = \frac{- ( - 3 ) - 15}{2 \cdot 18} : - \frac{1}{3}

\frac{- ( - 3 ) - 15}{2 \cdot 18}

Apply rule

- (- a) = a

= \frac{3 - 15}{2 \cdot 18}

Subtract the numbers:

3 - 15 = - 12

= \frac{- 12}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{- 12}{36}

Apply the fraction rule:

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

= - \frac{12}{36}

Cancel the common factor:

12

= - \frac{1}{3}

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = - \frac{1}{3}

Substitute back

u = tan (x)

tan (x) = \frac{1}{2}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{2}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{2} : x = arctan (\frac{1}{2}) + πn

tan (x) = \frac{1}{2}

Apply trig inverse properties

tan (x) = \frac{1}{2}

General solutions for

tan (x) = \frac{1}{2}

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

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

tan (x) = - \frac{1}{3}

Apply trig inverse properties

tan (x) = - \frac{1}{3}

General solutions for

tan (x) = - \frac{1}{3}

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

x = arctan (- \frac{1}{3}) + πn

x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = arctan (\frac{1}{2}) + πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 0.46364 \dots + πn, x = - 0.32175 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 18sec^2(x)-3tan(x)=21 ?
The general solution for 18sec^2(x)-3tan(x)=21 is x=0.46364…+pin,x=-0.32175…+pin