What is the general solution for 2sin(x)=tan(x)cos(x)-cos(x) ?

The general solution for 2sin(x)=tan(x)cos(x)-cos(x) is x=(3pi)/4+pin

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2sin(x)=tan(x)cos(x)-cos(x)

2 sin (x) = tan (x) cos (x) - cos (x)

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

2 sin (x) = tan (x) cos (x) - cos (x)

Subtract

tan (x) cos (x) - cos (x)

from both sides

2 sin (x) - tan (x) cos (x) + cos (x) = 0

Rewrite using trig identities

cos (x) + 2 sin (x) - cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

cos (x) + 2 sin (x) - cos (x) \frac{sin ( x )}{cos ( x )} : cos (x) + sin (x)

cos (x) + 2 sin (x) - cos (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= cos (x) + 2 sin (x) - sin (x)

Add similar elements:

2 sin (x) - sin (x) = sin (x)

= cos (x) + sin (x)

= cos (x) + sin (x)

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 + tan (x) = 0

1 + tan (x) = 0

Move

1

to the right side

1 + tan (x) = 0

Subtract

1

from both sides

1 + tan (x) - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x, z = 3 - sin (2 x + y)

tan (x) = \frac{27}{48}

cos (θ) - sin (θ) = 1, 0 \leq θ \leq 2 π

2 cos (x) = - 0.2

cot (θ - \frac{π}{2}) = 1.07

What is the general solution for 2sin(x)=tan(x)cos(x)-cos(x) ?
The general solution for 2sin(x)=tan(x)cos(x)-cos(x) is x=(3pi)/4+pin