What is the general solution for 7tan(4x+199)=8sin(4x+199) ?

The general solution for 7tan(4x+199)=8sin(4x+199) is

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7tan(4x+199)=8sin(4x+199)

Solution

7 tan (4 x + 199) = 8 sin (4 x + 199)

Solution

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

Degrees

x = - 2850.46503 \dots^{\circ} + 9 0^{\circ} n, x = - 2805.46503 \dots^{\circ} + 9 0^{\circ} n, x = - 2843.22627 \dots^{\circ} + 9 0^{\circ} n, x = - 2767.70378 \dots^{\circ} + 9 0^{\circ} n

Solution steps

7 tan (4 x + 199) = 8 sin (4 x + 199)

Subtract

8 sin (4 x + 199)

from both sides

7 tan (4 x + 199) - 8 sin (4 x + 199) = 0

Express with sin, cos

7 tan (199 + 4 x) - 8 sin (199 + 4 x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x)

Simplify

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x) : \frac{7 sin ( 199 + 4 x ) - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x)

Multiply

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} : \frac{7 sin ( 4 x + 199 )}{cos ( 199 + 4 x )}

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )}

Multiply fractions:

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

= \frac{sin ( 199 + 4 x ) \cdot 7}{cos ( 199 + 4 x )}

= \frac{7 sin ( 4 x + 199 )}{cos ( 4 x + 199 )} - 8 sin (4 x + 199)

Convert element to fraction:

8 sin (4 x + 199) = \frac{8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

= \frac{sin ( 199 + 4 x ) \cdot 7}{cos ( 199 + 4 x )} - \frac{8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( 199 + 4 x ) \cdot 7 - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

= \frac{7 sin ( 199 + 4 x ) - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

\frac{7 sin ( 199 + 4 x ) - 8 cos ( 199 + 4 x ) sin ( 199 + 4 x )}{cos ( 199 + 4 x )} = 0

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

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) = 0

Factor

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) : - sin (199 + 4 x) (8 cos (199 + 4 x) - 7)

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x)

Factor out common term

- sin (199 + 4 x)

= - sin (199 + 4 x) (- 7 + 8 cos (199 + 4 x))

- sin (199 + 4 x) (8 cos (199 + 4 x) - 7) = 0

Solving each part separately

sin (199 + 4 x) = 0 or 8 cos (199 + 4 x) - 7 = 0

sin (199 + 4 x) = 0 : x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

sin (199 + 4 x) = 0

General solutions for

sin (199 + 4 x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

199 + 4 x = 0 + 2 πn, 199 + 4 x = π + 2 πn

199 + 4 x = 0 + 2 πn, 199 + 4 x = π + 2 πn

Solve

199 + 4 x = 0 + 2 πn : x = \frac{πn}{2} - \frac{199}{4}

199 + 4 x = 0 + 2 πn

0 + 2 πn = 2 πn

199 + 4 x = 2 πn

Move

199

to the right side

199 + 4 x = 2 πn

Subtract

199

from both sides

199 + 4 x - 199 = 2 πn - 199

Simplify

4 x = 2 πn - 199

4 x = 2 πn - 199

Divide both sides by

4

4 x = 2 πn - 199

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4} - \frac{199}{4}

Simplify

x = \frac{πn}{2} - \frac{199}{4}

x = \frac{πn}{2} - \frac{199}{4}

Solve

199 + 4 x = π + 2 πn : x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

199 + 4 x = π + 2 πn

Move

199

to the right side

199 + 4 x = π + 2 πn

Subtract

199

from both sides

199 + 4 x - 199 = π + 2 πn - 199

Simplify

4 x = π + 2 πn - 199

4 x = π + 2 πn - 199

Divide both sides by

4

4 x = π + 2 πn - 199

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4} : \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

\frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

Group like terms

= \frac{π}{4} - \frac{199}{4} + \frac{2 πn}{4}

Cancel

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

8 cos (199 + 4 x) - 7 = 0 : x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

8 cos (199 + 4 x) - 7 = 0

Move

7

to the right side

8 cos (199 + 4 x) - 7 = 0

Add

7

to both sides

8 cos (199 + 4 x) - 7 + 7 = 0 + 7

Simplify

8 cos (199 + 4 x) = 7

8 cos (199 + 4 x) = 7

Divide both sides by

8

8 cos (199 + 4 x) = 7

Divide both sides by

8

\frac{8 cos ( 199 + 4 x )}{8} = \frac{7}{8}

Simplify

cos (199 + 4 x) = \frac{7}{8}

cos (199 + 4 x) = \frac{7}{8}

Apply trig inverse properties

cos (199 + 4 x) = \frac{7}{8}

General solutions for

cos (199 + 4 x) = \frac{7}{8}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

199 + 4 x = arccos (\frac{7}{8}) + 2 πn, 199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

199 + 4 x = arccos (\frac{7}{8}) + 2 πn, 199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

Solve

199 + 4 x = arccos (\frac{7}{8}) + 2 πn : x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

199 + 4 x = arccos (\frac{7}{8}) + 2 πn

Move

199

to the right side

199 + 4 x = arccos (\frac{7}{8}) + 2 πn

Subtract

199

from both sides

199 + 4 x - 199 = arccos (\frac{7}{8}) + 2 πn - 199

Simplify

4 x = arccos (\frac{7}{8}) + 2 πn - 199

4 x = arccos (\frac{7}{8}) + 2 πn - 199

Divide both sides by

4

4 x = arccos (\frac{7}{8}) + 2 πn - 199

Divide both sides by

4

\frac{4 x}{4} = \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} = \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4} : - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

\frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Group like terms

= - \frac{199}{4} + \frac{2 πn}{4} + \frac{arccos ( \frac{7}{8} )}{4}

Cancel

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

Solve

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn : x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

Move

199

to the right side

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

Subtract

199

from both sides

199 + 4 x - 199 = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

Simplify

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

Divide both sides by

4

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

Divide both sides by

4

\frac{4 x}{4} = \frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} = \frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4} : \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

\frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

Group like terms

= \frac{2 π}{4} - \frac{199}{4} + \frac{2 πn}{4} - \frac{arccos ( \frac{7}{8} )}{4}

Cancel

\frac{2 π}{4} : \frac{π}{2}

\frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} - \frac{199}{4} + \frac{2 πn}{4} - \frac{arccos ( \frac{7}{8} )}{4}

Cancel

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

Combine all the solutions

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

Show solutions in decimal form

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for 7tan(4x+199)=8sin(4x+199) ?
The general solution for 7tan(4x+199)=8sin(4x+199) is