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Popular >

csc(x)-sin(x)=cos(x)cot(3x-50)

Solution

csc (x) - sin (x) = cos (x) cot (3 x - 50)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 1432.39448 \dots^{\circ} + 18 0^{\circ} n, x = 1522.39448 \dots^{\circ} + 18 0^{\circ} n

Solution steps

csc (x) - sin (x) = cos (x) cot (3 x - 50)

Subtract

cos (x) cot (3 x - 50)

from both sides

csc (x) - sin (x) - cos (x) cot (3 x - 50) = 0

Express with sin, cos

csc (x) - sin (x) - cos (x) cot (- 50 + 3 x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} - sin (x) - cos (x) cot (- 50 + 3 x)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

Simplify

\frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )} : \frac{sin ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

\frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

Multiply

cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )} : \frac{cos ( 3 x - 50 ) cos ( x )}{sin ( - 50 + 3 x )}

cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

Multiply fractions:

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

= \frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )}

= \frac{1}{sin ( x )} - sin (x) - \frac{cos ( 3 x - 50 ) cos ( x )}{sin ( 3 x - 50 )}

Convert element to fraction:

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

= \frac{1}{sin ( x )} - \frac{sin ( x )}{1} - \frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )}

Least Common Multiplier of

sin (x), 1, sin (- 50 + 3 x) : sin (x) sin (3 x - 50)

sin (x), 1, sin (- 50 + 3 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin (x) sin (3 x - 50)

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) sin (3 x - 50)

For

\frac{1}{sin ( x )} :

multiply the denominator and numerator by

sin (3 x - 50)

\frac{1}{sin ( x )} = \frac{1 \cdot sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} = \frac{sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )}

For

\frac{sin ( x )}{1} :

multiply the denominator and numerator by

sin (x) sin (3 x - 50)

\frac{sin ( x )}{1} = \frac{sin ( x ) sin ( x ) sin ( 3 x - 50 )}{1 \cdot sin ( x ) sin ( 3 x - 50 )} = \frac{sin ^{2} ( x ) sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )}

For

\frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )} :

multiply the denominator and numerator by

sin (x)

\frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )} = \frac{cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )}

= \frac{sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} - \frac{sin ^{2} ( x ) sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} - \frac{cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 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 ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

= \frac{sin ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

\frac{sin ( - 50 + 3 x ) - sin ( - 50 + 3 x ) sin ^{2} ( x ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )} = 0

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

sin (- 50 + 3 x) - sin (- 50 + 3 x) sin^{2} (x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

Rewrite using trig identities

sin (- 50 + 3 x) - sin (- 50 + 3 x) sin^{2} (x) - cos (- 50 + 3 x) cos (x) sin (x)

Use the Pythagorean identity:

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

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

= sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x)

Simplify

sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x) : cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x)

Expand

- sin (- 50 + 3 x) (1 - cos^{2} (x)) : - sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

- sin (- 50 + 3 x) (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - sin (- 50 + 3 x), b = 1, c = cos^{2} (x)

= - sin (- 50 + 3 x) \cdot 1 - (- sin (- 50 + 3 x)) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 1 \cdot sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

Multiply:

1 \cdot sin (- 50 + 3 x) = sin (- 50 + 3 x)

= - sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

= sin (- 50 + 3 x) - sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

Add similar elements:

sin (3 x - 50) - sin (3 x - 50) = 0

= cos^{2} (x) sin (3 x - 50) - cos (3 x - 50) cos (x) sin (x)

= cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

Factor

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x) : cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x))

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

Apply exponent rule:

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

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

= cos (x) cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

Factor out common term

cos (x)

= cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x))

cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x)) = 0

Solving each part separately

cos (x) = 0 or cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0 : x = πn + 25, x = \frac{π}{2} + 25 + πn

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0

Rewrite using trig identities

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (- 50 + 3 x - x)

sin (- 50 + 3 x - x) = 0

General solutions for

sin (- 50 + 3 x - 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}

- 50 + 3 x - x = 0 + 2 πn, - 50 + 3 x - x = π + 2 πn

- 50 + 3 x - x = 0 + 2 πn, - 50 + 3 x - x = π + 2 πn

Solve

- 50 + 3 x - x = 0 + 2 πn : x = πn + 25

- 50 + 3 x - x = 0 + 2 πn

Add similar elements:

3 x - x = 2 x

- 50 + 2 x = 0 + 2 πn

0 + 2 πn = 2 πn

- 50 + 2 x = 2 πn

Move

50

to the right side

- 50 + 2 x = 2 πn

Add

50

to both sides

- 50 + 2 x + 50 = 2 πn + 50

Simplify

2 x = 2 πn + 50

2 x = 2 πn + 50

Divide both sides by

2

2 x = 2 πn + 50

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{50}{2} : πn + 25

\frac{2 πn}{2} + \frac{50}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn + \frac{50}{2}

Divide the numbers:

\frac{50}{2} = 25

= πn + 25

x = πn + 25

x = πn + 25

x = πn + 25

Solve

- 50 + 3 x - x = π + 2 πn : x = \frac{π}{2} + 25 + πn

- 50 + 3 x - x = π + 2 πn

Add similar elements:

3 x - x = 2 x

- 50 + 2 x = π + 2 πn

Move

50

to the right side

- 50 + 2 x = π + 2 πn

Add

50

to both sides

- 50 + 2 x + 50 = π + 2 πn + 50

Simplify

2 x = π + 2 πn + 50

2 x = π + 2 πn + 50

Divide both sides by

2

2 x = π + 2 πn + 50

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2} : \frac{π}{2} + 25 + πn

\frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

Group like terms

= \frac{π}{2} + \frac{50}{2} + \frac{2 πn}{2}

Divide the numbers:

\frac{50}{2} = 25

= \frac{π}{2} + 25 + \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = πn + 25, x = \frac{π}{2} + 25 + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

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