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Popular Trigonometry >

solvefor x,cot(-x)cos(-x)+sin(-x)=-1/f

Solution

solve for

x, cot (- x) cos (- x) + sin (- x) = - \frac{1}{f}

Solution

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

Solution steps

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

Rewrite using trig identities

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

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

Simplify

- sin (x) + cos (x) (- cot (x)) : - sin (x) - cos (x) cot (x)

- sin (x) + cos (x) (- cot (x))

Remove parentheses:

(- a) = - a

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

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

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

Subtract

- \frac{1}{f}

from both sides

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

Simplify

- sin (x) - cos (x) cot (x) + \frac{1}{f} : \frac{- f sin ( x ) - f cos ( x ) cot ( x ) + 1}{f}

- sin (x) - cos (x) cot (x) + \frac{1}{f}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

- f sin (x) - f cos (x) cot (x) + 1 = 0

Rewrite using trig identities

1 - sin (x) f - cos (x) cot (x) f

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

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

cos (x) cos (x) f

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= cos^{2} (x) f

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

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

Use the Pythagorean identity:

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

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

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

Combine the fractions

- \frac{f ( - sin ^{2} ( x ) + 1 )}{sin ( x )} - f sin (x) : - \frac{f}{sin ( x )}

- \frac{f ( - sin ^{2} ( x ) + 1 )}{sin ( x )} - f sin (x)

Convert element to fraction:

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

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

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

- (1 - sin^{2} (x)) f - sin (x) f sin (x)

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

sin (x) f sin (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= f sin^{2} (x)

= - f (- sin^{2} (x) + 1) - f sin^{2} (x)

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

Expand

- (1 - sin^{2} (x)) f - f sin^{2} (x) : - f

- (1 - sin^{2} (x)) f - f sin^{2} (x)

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

Expand

- f (1 - sin^{2} (x)) : - f + f sin^{2} (x)

- f (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - f, b = 1, c = sin^{2} (x)

= - f \cdot 1 - (- f) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 1 \cdot f + f sin^{2} (x)

Multiply:

1 \cdot f = f

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

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

Add similar elements:

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

= - f

= \frac{- f}{sin ( x )}

Apply the fraction rule:

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

= - \frac{f}{sin ( x )}

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

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

Multiply both sides by

sin (x)

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

Multiply both sides by

sin (x)

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

Simplify

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

Simplify

1 \cdot sin (x) : sin (x)

1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= sin (x)

Simplify

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

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

Multiply fractions:

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

= - \frac{f sin ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= - f

Simplify

0 \cdot sin (x) : 0

0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

sin (x) - f = 0

sin (x) - f = 0

sin (x) - f = 0

Move

f

to the right side

sin (x) - f = 0

Add

f

to both sides

sin (x) - f + f = 0 + f

Simplify

sin (x) = f

sin (x) = f

Apply trig inverse properties

sin (x) = f

General solutions for

sin (x) = f

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

Graph

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