What is the general solution for sin(x)=(-(6))/((7)cos(x)) ?

The general solution for sin(x)=(-(6))/((7)cos(x)) is No Solution for x\in\mathbb{R}

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Popular >

sin(x)=(-(6))/((7)cos(x))

Solution

sin (x) = \frac{- ( 6 )}{( 7 ) cos ( x )}

Solution

No Solution for x \in R

Solution steps

sin (x) = \frac{- ( 6 )}{( 7 ) cos ( x )}

Subtract

\frac{- 6}{7 cos ( x )}

from both sides

sin (x) + \frac{6}{7 cos ( x )} = 0

Simplify

sin (x) + \frac{6}{7 cos ( x )} : \frac{7 sin ( x ) cos ( x ) + 6}{7 cos ( x )}

sin (x) + \frac{6}{7 cos ( x )}

Convert element to fraction:

sin (x) = \frac{sin ( x ) 7 cos ( x )}{7 cos ( x )}

= \frac{sin ( x ) \cdot 7 cos ( x )}{7 cos ( x )} + \frac{6}{7 cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 7 cos ( x ) + 6}{7 cos ( x )}

\frac{7 sin ( x ) cos ( x ) + 6}{7 cos ( x )} = 0

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

7 sin (x) cos (x) + 6 = 0

Rewrite using trig identities

7 sin (x) cos (x) + 6

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 6 + 7 \cdot \frac{sin ( 2 x )}{2}

6 + 7 \cdot \frac{sin ( 2 x )}{2} = 0

Move

6

to the right side

6 + 7 \cdot \frac{sin ( 2 x )}{2} = 0

Subtract

6

from both sides

6 + 7 \cdot \frac{sin ( 2 x )}{2} - 6 = 0 - 6

Simplify

7 \cdot \frac{sin ( 2 x )}{2} = - 6

7 \cdot \frac{sin ( 2 x )}{2} = - 6

Refine

7 \cdot \frac{sin ( 2 x )}{2} : \frac{7 sin ( 2 x )}{2}

7 \cdot \frac{sin ( 2 x )}{2}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 7}{2}

\frac{7 sin ( 2 x )}{2} = - 6

Multiply both sides by

2

\frac{7 sin ( 2 x )}{2} = - 6

Multiply both sides by

2

\frac{7 sin ( 2 x )}{2} \cdot 2 = - 6 \cdot 2

Simplify

\frac{7 sin ( 2 x )}{2} \cdot 2 = - 6 \cdot 2

Simplify

\frac{7 sin ( 2 x )}{2} \cdot 2 : 7 sin (2 x)

\frac{7 sin ( 2 x )}{2} \cdot 2

Multiply fractions:

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

= \frac{7 sin ( 2 x ) \cdot 2}{2}

Cancel the common factor:

2

= 7 sin (2 x)

Simplify

- 6 \cdot 2 : - 12

- 6 \cdot 2

Multiply the numbers:

6 \cdot 2 = 12

= - 12

7 sin (2 x) = - 12

7 sin (2 x) = - 12

7 sin (2 x) = - 12

Divide both sides by

7

7 sin (2 x) = - 12

Divide both sides by

7

\frac{7 sin ( 2 x )}{7} = \frac{- 12}{7}

Simplify

sin (2 x) = - \frac{12}{7}

sin (2 x) = - \frac{12}{7}

- 1 \leq sin (x) \leq 1

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=(-(6))/((7)cos(x)) ?
The general solution for sin(x)=(-(6))/((7)cos(x)) is No Solution for x\in\mathbb{R}