What is the general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) ?

The general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) is x=1.10714…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

x = 1.10714 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

\frac{1}{2} sin^{2} (x)

from both sides

cot (x) (1 - cos^{2} (x)) - \frac{1}{2} sin^{2} (x) = 0

Simplify

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

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

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

\frac{1}{2} sin^{2} (x)

Multiply fractions:

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

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

Multiply:

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

\frac{2 cot ( x ) ( 1 - cos ^{2} ( x ) ) - sin ^{2} ( x )}{2} = 0

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

2 cot (x) (1 - cos^{2} (x)) - sin^{2} (x) = 0

Rewrite using trig identities

- sin^{2} (x) + (1 - cos^{2} (x)) \cdot 2 cot (x)

Use the Pythagorean identity:

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

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

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

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

Factor

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

- sin^{2} (x) + 2 cot (x) sin^{2} (x)

Factor out common term

sin^{2} (x)

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

sin^{2} (x) (2 cot (x) - 1) = 0

Solving each part separately

sin^{2} (x) = 0 or 2 cot (x) - 1 = 0

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

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

2 cot (x) - 1 = 0 : x = arccot (\frac{1}{2}) + πn

2 cot (x) - 1 = 0

Move

1

to the right side

2 cot (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 cot (x) = 1

2 cot (x) = 1

Divide both sides by

2

2 cot (x) = 1

Divide both sides by

2

\frac{2 cot ( x )}{2} = \frac{1}{2}

Simplify

cot (x) = \frac{1}{2}

cot (x) = \frac{1}{2}

Apply trig inverse properties

cot (x) = \frac{1}{2}

General solutions for

cot (x) = \frac{1}{2}

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccot (\frac{1}{2}) + πn

Since the equation is undefined for:

2 πn, π + 2 πn

x = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn

Popular Examples

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

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

cos(x)= 20/29

cos (x) = \frac{20}{29}

2cot^4(x)-cot^2(x)-15=0

2 cot^{4} (x) - cot^{2} (x) - 15 = 0

cos(θ)=(sqrt(11))/6

cos (θ) = \frac{11}{6}

1+cos(x)=3cos(x)

1 + cos (x) = 3 cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) ?
The general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) is x=1.10714…+pin