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

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

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Popular >

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

2 cos (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

sin (x) - 2 cos (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)

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

Add similar elements:

sin (x) + sin (x) = 2 sin (x)

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

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

Divide both sides by

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

\frac{- 2 cos ( x ) + 2 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- 2 + \frac{2 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

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

- 2 + 2 tan (x) = 0

- 2 + 2 tan (x) = 0

Move

2

to the right side

- 2 + 2 tan (x) = 0

Add

2

to both sides

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

Simplify

2 tan (x) = 2

2 tan (x) = 2

Divide both sides by

2

2 tan (x) = 2

Divide both sides by

2

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

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{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

cos(x)+sin(x)=(sqrt(6))/2

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

6tan^2(x)-2=0

6 tan^{2} (x) - 2 = 0

tan(x)=infinity

tan (x) = \infty

2arctanh(((x-2))/((x+1)))=ln(2)

2 arctanh (\frac{( x - 2 )}{( x + 1 )}) = ln (2)

tan(x)=-1.5

tan (x) = - 1.5

Frequently Asked Questions (FAQ)

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