What is the general solution for ((3))/((cos(x)))=8-5tan(x) ?

The general solution for ((3))/((cos(x)))=8-5tan(x) is x=0.68857…+2pin,x=pi+1.33581…+2pin

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((3))/((cos(x)))=8-5tan(x)

Solution

\frac{( 3 )}{( cos ( x ) )} = 8 - 5 tan (x)

Solution

x = 0.68857 \dots + 2 πn, x = π + 1.33581 \dots + 2 πn

Degrees

x = 39.45263 \dots^{\circ} + 36 0^{\circ} n, x = 256.53659 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{( 3 )}{( cos ( x ) )} = 8 - 5 tan (x)

Subtract

8 - 5 tan (x)

from both sides

\frac{3}{cos ( x )} - 8 + 5 tan (x) = 0

Simplify

\frac{3}{cos ( x )} - 8 + 5 tan (x) : \frac{3 - 8 cos ( x ) + 5 tan ( x ) cos ( x )}{cos ( x )}

\frac{3}{cos ( x )} - 8 + 5 tan (x)

Convert element to fraction:

8 = \frac{8 cos ( x )}{cos ( x )}, 5 tan (x) = \frac{5 tan ( x ) cos ( x )}{cos ( x )}

= \frac{3}{cos ( x )} - \frac{8 cos ( x )}{cos ( x )} + \frac{5 tan ( x ) cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{3 - 8 cos ( x ) + 5 tan ( x ) cos ( x )}{cos ( x )}

\frac{3 - 8 cos ( x ) + 5 tan ( x ) cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 - 8 cos (x) + 5 tan (x) cos (x) = 0

Express with sin, cos

3 - 8 cos (x) + 5 \cdot \frac{sin ( x )}{cos ( x )} cos (x) = 0

Simplify

3 - 8 cos (x) + 5 \cdot \frac{sin ( x )}{cos ( x )} cos (x) : 3 - 8 cos (x) + 5 sin (x)

3 - 8 cos (x) + 5 \cdot \frac{sin ( x )}{cos ( x )} cos (x)

5 \cdot \frac{sin ( x )}{cos ( x )} cos (x) = 5 sin (x)

5 \cdot \frac{sin ( x )}{cos ( x )} cos (x)

Multiply fractions:

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

= \frac{sin ( x ) \cdot 5 cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= sin (x) \cdot 5

= 3 - 8 cos (x) + 5 sin (x)

3 - 8 cos (x) + 5 sin (x) = 0

Add

8 cos (x)

to both sides

5 sin (x) + 3 = 8 cos (x)

Square both sides

(5 sin (x) + 3)^{2} = (8 cos (x))^{2}

Subtract

(8 cos (x))^{2}

from both sides

(5 sin (x) + 3)^{2} - 64 cos^{2} (x) = 0

Rewrite using trig identities

(3 + 5 sin (x))^{2} - 64 cos^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= (3 + 5 sin (x))^{2} - 64 (1 - sin^{2} (x))

Simplify

(3 + 5 sin (x))^{2} - 64 (1 - sin^{2} (x)) : 89 sin^{2} (x) + 30 sin (x) - 55

(3 + 5 sin (x))^{2} - 64 (1 - sin^{2} (x))

(3 + 5 sin (x))^{2} : 9 + 30 sin (x) + 25 sin^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 3, b = 5 sin (x)

= 3^{2} + 2 \cdot 3 \cdot 5 sin (x) + (5 sin (x))^{2}

Simplify

3^{2} + 2 \cdot 3 \cdot 5 sin (x) + (5 sin (x))^{2} : 9 + 30 sin (x) + 25 sin^{2} (x)

3^{2} + 2 \cdot 3 \cdot 5 sin (x) + (5 sin (x))^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

2 \cdot 3 \cdot 5 sin (x) = 30 sin (x)

2 \cdot 3 \cdot 5 sin (x)

Multiply the numbers:

2 \cdot 3 \cdot 5 = 30

= 30 sin (x)

(5 sin (x))^{2} = 25 sin^{2} (x)

(5 sin (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 5^{2} sin^{2} (x)

5^{2} = 25

= 25 sin^{2} (x)

= 9 + 30 sin (x) + 25 sin^{2} (x)

= 9 + 30 sin (x) + 25 sin^{2} (x)

= 9 + 30 sin (x) + 25 sin^{2} (x) - 64 (1 - sin^{2} (x))

Expand

- 64 (1 - sin^{2} (x)) : - 64 + 64 sin^{2} (x)

- 64 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 64, b = 1, c = sin^{2} (x)

= - 64 \cdot 1 - (- 64) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 64 \cdot 1 + 64 sin^{2} (x)

Multiply the numbers:

64 \cdot 1 = 64

= - 64 + 64 sin^{2} (x)

= 9 + 30 sin (x) + 25 sin^{2} (x) - 64 + 64 sin^{2} (x)

Simplify

9 + 30 sin (x) + 25 sin^{2} (x) - 64 + 64 sin^{2} (x) : 89 sin^{2} (x) + 30 sin (x) - 55

9 + 30 sin (x) + 25 sin^{2} (x) - 64 + 64 sin^{2} (x)

Group like terms

= 30 sin (x) + 25 sin^{2} (x) + 64 sin^{2} (x) + 9 - 64

Add similar elements:

25 sin^{2} (x) + 64 sin^{2} (x) = 89 sin^{2} (x)

= 30 sin (x) + 89 sin^{2} (x) + 9 - 64

Add/Subtract the numbers:

9 - 64 = - 55

= 89 sin^{2} (x) + 30 sin (x) - 55

= 89 sin^{2} (x) + 30 sin (x) - 55

= 89 sin^{2} (x) + 30 sin (x) - 55

- 55 + 30 sin (x) + 89 sin^{2} (x) = 0

Solve by substitution

- 55 + 30 sin (x) + 89 sin^{2} (x) = 0

Let:

sin (x) = u

- 55 + 30 u + 89 u^{2} = 0

- 55 + 30 u + 89 u^{2} = 0 : u = \frac{- 15 + 32 5}{89}, u = - \frac{15 + 32 5}{89}

- 55 + 30 u + 89 u^{2} = 0

Write in the standard form

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

89 u^{2} + 30 u - 55 = 0

Solve with the quadratic formula

89 u^{2} + 30 u - 55 = 0

Quadratic Equation Formula:

For

a = 89, b = 30, c = - 55

u_{1, 2} = \frac{- 30 \pm 3 0 ^{2} - 4 \cdot 89 ( - 55 )}{2 \cdot 89}

u_{1, 2} = \frac{- 30 \pm 3 0 ^{2} - 4 \cdot 89 ( - 55 )}{2 \cdot 89}

3 0^{2} - 4 \cdot 89 (- 55) = 645

3 0^{2} - 4 \cdot 89 (- 55)

Apply rule

- (- a) = a

= 3 0^{2} + 4 \cdot 89 \cdot 55

Multiply the numbers:

4 \cdot 89 \cdot 55 = 19580

= 3 0^{2} + 19580

3 0^{2} = 900

= 900 + 19580

Add the numbers:

900 + 19580 = 20480

= 20480

Prime factorization of

20480 : 2^{12} \cdot 5

20480

20480

divides by

220480 = 10240 \cdot 2

= 2 \cdot 10240

10240

divides by

210240 = 5120 \cdot 2

= 2 \cdot 2 \cdot 5120

5120

divides by

25120 = 2560 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2560

2560

divides by

22560 = 1280 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 1280

1280

divides by

21280 = 640 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 640

640

divides by

2640 = 320 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 320

320

divides by

2320 = 160 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 160

160

divides by

2160 = 80 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 80

80

divides by

280 = 40 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

= 2^{12} \cdot 5

= 2^{12} \cdot 5

Apply radical rule:

= 5 2^{12}

Apply radical rule:

2^{12} = 2^{\frac{12}{2}} = 2^{6}

= 2^{6} 5

Refine

= 645

u_{1, 2} = \frac{- 30 \pm 64 5}{2 \cdot 89}

Separate the solutions

u_{1} = \frac{- 30 + 64 5}{2 \cdot 89}, u_{2} = \frac{- 30 - 64 5}{2 \cdot 89}

u = \frac{- 30 + 64 5}{2 \cdot 89} : \frac{- 15 + 32 5}{89}

\frac{- 30 + 64 5}{2 \cdot 89}

Multiply the numbers:

2 \cdot 89 = 178

= \frac{- 30 + 64 5}{178}

Factor

- 30 + 645 : 2 (- 15 + 325)

- 30 + 645

Rewrite as

= - 2 \cdot 15 + 2 \cdot 325

Factor out common term

2

= 2 (- 15 + 325)

= \frac{2 ( - 15 + 32 5 )}{178}

Cancel the common factor:

2

= \frac{- 15 + 32 5}{89}

u = \frac{- 30 - 64 5}{2 \cdot 89} : - \frac{15 + 32 5}{89}

\frac{- 30 - 64 5}{2 \cdot 89}

Multiply the numbers:

2 \cdot 89 = 178

= \frac{- 30 - 64 5}{178}

Factor

- 30 - 645 : - 2 (15 + 325)

- 30 - 645

Rewrite as

= - 2 \cdot 15 - 2 \cdot 325

Factor out common term

2

= - 2 (15 + 325)

= - \frac{2 ( 15 + 32 5 )}{178}

Cancel the common factor:

2

= - \frac{15 + 32 5}{89}

The solutions to the quadratic equation are:

u = \frac{- 15 + 32 5}{89}, u = - \frac{15 + 32 5}{89}

Substitute back

u = sin (x)

sin (x) = \frac{- 15 + 32 5}{89}, sin (x) = - \frac{15 + 32 5}{89}

sin (x) = \frac{- 15 + 32 5}{89}, sin (x) = - \frac{15 + 32 5}{89}

sin (x) = \frac{- 15 + 32 5}{89} : x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

sin (x) = \frac{- 15 + 32 5}{89}

Apply trig inverse properties

sin (x) = \frac{- 15 + 32 5}{89}

General solutions for

sin (x) = \frac{- 15 + 32 5}{89}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

sin (x) = - \frac{15 + 32 5}{89} : x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

sin (x) = - \frac{15 + 32 5}{89}

Apply trig inverse properties

sin (x) = - \frac{15 + 32 5}{89}

General solutions for

sin (x) = - \frac{15 + 32 5}{89}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

\frac{3}{cos ( x )} = 8 - 5 tan (x)

Remove the ones that don't agree with the equation.

Check the solution

arcsin (\frac{- 15 + 32 5}{89}) + 2 πn :

True

arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

Plug in

n = 1

arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1

For

\frac{3}{cos ( x )} = 8 - 5 tan (x)

plug in

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1

\frac{3}{cos ( arcsin ( \frac{- 15 + 32 5}{89} ) + 2 π 1 )} = 8 - 5 tan (arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1)

Refine

3.88525 \dots = 3.88525 \dots

\Rightarrow True

Check the solution

π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn :

False

π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1

For

\frac{3}{cos ( x )} = 8 - 5 tan (x)

plug in

x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1

\frac{3}{cos ( π - arcsin ( \frac{- 15 + 32 5}{89} ) + 2 π 1 )} = 8 - 5 tan (π - arcsin (\frac{- 15 + 32 5}{89}) + 2 π 1)

Refine

- 3.88525 \dots = 12.11474 \dots

\Rightarrow False

Check the solution

arcsin (- \frac{15 + 32 5}{89}) + 2 πn :

False

arcsin (- \frac{15 + 32 5}{89}) + 2 πn

Plug in

n = 1

arcsin (- \frac{15 + 32 5}{89}) + 2 π 1

For

\frac{3}{cos ( x )} = 8 - 5 tan (x)

plug in

x = arcsin (- \frac{15 + 32 5}{89}) + 2 π 1

\frac{3}{cos ( arcsin ( - \frac{15 + 32 5}{89} ) + 2 π 1 )} = 8 - 5 tan (arcsin (- \frac{15 + 32 5}{89}) + 2 π 1)

Refine

12.88525 \dots = 28.88525 \dots

\Rightarrow False

Check the solution

π + arcsin (\frac{15 + 32 5}{89}) + 2 πn :

True

π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{15 + 32 5}{89}) + 2 π 1

For

\frac{3}{cos ( x )} = 8 - 5 tan (x)

plug in

x = π + arcsin (\frac{15 + 32 5}{89}) + 2 π 1

\frac{3}{cos ( π + arcsin ( \frac{15 + 32 5}{89} ) + 2 π 1 )} = 8 - 5 tan (π + arcsin (\frac{15 + 32 5}{89}) + 2 π 1)

Refine

- 12.88525 \dots = - 12.88525 \dots

\Rightarrow True

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Show solutions in decimal form

x = 0.68857 \dots + 2 πn, x = π + 1.33581 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for ((3))/((cos(x)))=8-5tan(x) ?
The general solution for ((3))/((cos(x)))=8-5tan(x) is x=0.68857…+2pin,x=pi+1.33581…+2pin