What is the general solution for sin(x)+cot(x)cos(x)=sqrt(3) ?

The general solution for sin(x)+cot(x)cos(x)=sqrt(3) is x=0.61547…+2pin,x=pi-0.61547…+2pin

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sin(x)+cot(x)cos(x)=sqrt(3)

Solution

sin (x) + cot (x) cos (x) = 3

Solution

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

Degrees

x = 35.26438 \dots^{\circ} + 36 0^{\circ} n, x = 144.73561 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + cot (x) cos (x) = 3

Subtract

3

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

Convert element to fraction:

sin (x) = \frac{sin ( x ) sin ( x )}{sin ( x )}, 3 = \frac{3 sin ( x )}{sin ( x )}

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

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

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - 3 sin (x) + 1

- 3 sin (x) + 1 = 0

Move

1

to the right side

- 3 sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

- 3 sin (x) = - 1

- 3 sin (x) = - 1

Divide both sides by

- 3

- 3 sin (x) = - 1

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 sin ( x )}{- 3} : sin (x)

\frac{- 3 sin ( x )}{- 3}

Apply the fraction rule:

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

= \frac{3 sin ( x )}{3}

Cancel the common factor:

3

= sin (x)

Simplify

\frac{- 1}{- 3} : \frac{3}{3}

\frac{- 1}{- 3}

Apply the fraction rule:

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

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

sin (x) = \frac{3}{3}

sin (x) = \frac{3}{3}

sin (x) = \frac{3}{3}

Apply trig inverse properties

sin (x) = \frac{3}{3}

General solutions for

sin (x) = \frac{3}{3}

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

x = arcsin (\frac{3}{3}) + 2 πn, x = π - arcsin (\frac{3}{3}) + 2 πn

x = arcsin (\frac{3}{3}) + 2 πn, x = π - arcsin (\frac{3}{3}) + 2 πn

Show solutions in decimal form

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

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cot(θ)= 1/(cot(θ))

cot (θ) = \frac{1}{cot ( θ )}

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2cot^2(x)+2csc^2(x)=1+4csc(x)

2 cot^{2} (x) + 2 csc^{2} (x) = 1 + 4 csc (x)

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+cot(x)cos(x)=sqrt(3) ?
The general solution for sin(x)+cot(x)cos(x)=sqrt(3) is x=0.61547…+2pin,x=pi-0.61547…+2pin