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

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

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Popular >

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

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Simplify

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

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

Join

tan (x) + \frac{cos ( x )}{sin ( x )} : \frac{tan ( x ) sin ( x ) + cos ( x )}{sin ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Multiply

cos (x) \frac{tan ( x ) sin ( x ) + cos ( x )}{sin ( x )} : \frac{cos ( x ) ( tan ( x ) sin ( x ) + cos ( x ) )}{sin ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = cos (x), b = cos (x), c = \frac{sin ^{2} ( x )}{cos ( x )}

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

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

Simplify

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

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

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin^{2} (x)

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

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

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

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

Use the Pythagorean identity:

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

= - 2 sin (x) + 1

- 2 sin (x) + 1 = 0

Move

1

to the right side

- 2 sin (x) + 1 = 0

Subtract

1

from both sides

- 2 sin (x) + 1 - 1 = 0 - 1

Simplify

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Divide both sides by

- 2

- 2 sin (x) = - 1

Divide both sides by

- 2

\frac{- 2 sin ( x )}{- 2} = \frac{- 1}{- 2}

Simplify

\frac{- 2 sin ( x )}{- 2} = \frac{- 1}{- 2}

Simplify

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

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Cancel the common factor:

2

= sin (x)

Simplify

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

General solutions for

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

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}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Popular Examples

2cos(2x)+sin(x)= 1/2

2 cos (2 x) + sin (x) = \frac{1}{2}

2cos^2(x)+cos(x)+1=0

2 cos^{2} (x) + cos (x) + 1 = 0

cos^2(x/2)-sin^2(x/2)=1-sin(x)

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

-2+3csc(x)=5

- 2 + 3 csc (x) = 5

sqrt(2)sin(2x)=sin(4x),0<= x<= pi

2 sin (2 x) = sin (4 x), 0 \leq x \leq π

Frequently Asked Questions (FAQ)

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