What is the general solution for (sin(x)+cos(x))/(sin(x))=(1+1)/(tan(x)) ?

The general solution for (sin(x)+cos(x))/(sin(x))=(1+1)/(tan(x)) is x= pi/4+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

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

Solution steps

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

Subtract

\frac{1 + 1}{tan ( x )}

from both sides

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

Simplify

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

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

Least Common Multiplier of

sin (x), tan (x) : sin (x) tan (x)

sin (x), tan (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

tan (x)

= sin (x) tan (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (x) tan (x)

For

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

multiply the denominator and numerator by

tan (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Convert element to fraction:

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

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

Expand

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

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

Expand

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

sin (x) (cos (x) + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = sin (x), b = cos (x), c = sin (x)

= sin (x) cos (x) + sin (x) sin (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)

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

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

Add similar elements:

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

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

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

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

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

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

sin^{2} (x) - cos (x) sin (x) = 0

Factor

sin^{2} (x) - cos (x) sin (x) : sin (x) (sin (x) - cos (x))

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

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

sin (x) (sin (x) - cos (x)) = 0

Solving each part separately

sin (x) = 0 or sin (x) - cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 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{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x)+cos(x))/(sin(x))=(1+1)/(tan(x)) ?
The general solution for (sin(x)+cos(x))/(sin(x))=(1+1)/(tan(x)) is x= pi/4+pin