What is the general solution for cot(x)-2tan(x)=-0.25 ?

The general solution for cot(x)-2tan(x)=-0.25 is x=0.65766…+pin,x=2.56707…+pin

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cot(x)-2tan(x)=-0.25

Solution

cot (x) - 2 tan (x) = - 0.25

Solution

x = 0.65766 \dots + πn, x = 2.56707 \dots + πn

Degrees

x = 37.68118 \dots^{\circ} + 18 0^{\circ} n, x = 147.08245 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cot (x) - 2 tan (x) = - 0.25

Subtract

- 0.25

from both sides

cot (x) - 2 tan (x) + 0.25 = 0

Rewrite using trig identities

0.25 + cot (x) - 2 tan (x)

Use the basic trigonometric identity:

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

= 0.25 + cot (x) - 2 \cdot \frac{1}{cot ( x )}

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

= 0.25 + cot (x) - \frac{2}{cot ( x )}

0.25 + cot (x) - \frac{2}{cot ( x )} = 0

Solve by substitution

0.25 + cot (x) - \frac{2}{cot ( x )} = 0

Let:

cot (x) = u

0.25 + u - \frac{2}{u} = 0

0.25 + u - \frac{2}{u} = 0 : u = \frac{- 1 + 129}{8}, u = - \frac{1 + 129}{8}

0.25 + u - \frac{2}{u} = 0

Multiply both sides by

u

0.25 + u - \frac{2}{u} = 0

Multiply both sides by

u

0.25 u + uu - \frac{2}{u} u = 0 \cdot u

Simplify

0.25 u + uu - \frac{2}{u} u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

- \frac{2}{u} u : - 2

- \frac{2}{u} u

Multiply fractions:

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

= - \frac{2 u}{u}

Cancel the common factor:

u

= - 2

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

0.25 u + u^{2} - 2 = 0

0.25 u + u^{2} - 2 = 0

0.25 u + u^{2} - 2 = 0

Solve

0.25 u + u^{2} - 2 = 0 : u = \frac{- 1 + 129}{8}, u = - \frac{1 + 129}{8}

0.25 u + u^{2} - 2 = 0

Multiply both sides by

100

0.25 u + u^{2} - 2 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.25 u \cdot 100 + u^{2} \cdot 100 - 2 \cdot 100 = 0 \cdot 100

Refine

25 u + 100 u^{2} - 200 = 0

25 u + 100 u^{2} - 200 = 0

Write in the standard form

a x^{2} + b x + c = 0

100 u^{2} + 25 u - 200 = 0

Solve with the quadratic formula

100 u^{2} + 25 u - 200 = 0

Quadratic Equation Formula:

For

a = 100, b = 25, c = - 200

u_{1, 2} = \frac{- 25 \pm 2 5 ^{2} - 4 \cdot 100 ( - 200 )}{2 \cdot 100}

u_{1, 2} = \frac{- 25 \pm 2 5 ^{2} - 4 \cdot 100 ( - 200 )}{2 \cdot 100}

2 5^{2} - 4 \cdot 100 (- 200) = 25129

2 5^{2} - 4 \cdot 100 (- 200)

Apply rule

- (- a) = a

= 2 5^{2} + 4 \cdot 100 \cdot 200

Multiply the numbers:

4 \cdot 100 \cdot 200 = 80000

= 2 5^{2} + 80000

2 5^{2} = 625

= 625 + 80000

Add the numbers:

625 + 80000 = 80625

= 80625

Prime factorization of

80625 : 3 \cdot 5^{4} \cdot 43

80625

80625

divides by

380625 = 26875 \cdot 3

= 3 \cdot 26875

26875

divides by

526875 = 5375 \cdot 5

= 3 \cdot 5 \cdot 5375

5375

divides by

55375 = 1075 \cdot 5

= 3 \cdot 5 \cdot 5 \cdot 1075

1075

divides by

51075 = 215 \cdot 5

= 3 \cdot 5 \cdot 5 \cdot 5 \cdot 215

215

divides by

5215 = 43 \cdot 5

= 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 43

3, 5, 43

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 43

= 3 \cdot 5^{4} \cdot 43

= 5^{4} \cdot 3 \cdot 43

Apply radical rule:

= 5^{4} 3 \cdot 43

Apply radical rule:

5^{4} = 5^{\frac{4}{2}} = 5^{2}

= 5^{2} 3 \cdot 43

Refine

= 25129

u_{1, 2} = \frac{- 25 \pm 25 129}{2 \cdot 100}

Separate the solutions

u_{1} = \frac{- 25 + 25 129}{2 \cdot 100}, u_{2} = \frac{- 25 - 25 129}{2 \cdot 100}

u = \frac{- 25 + 25 129}{2 \cdot 100} : \frac{- 1 + 129}{8}

\frac{- 25 + 25 129}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{- 25 + 25 129}{200}

Factor

- 25 + 25129 : 25 (- 1 + 129)

- 25 + 25129

Rewrite as

= - 25 \cdot 1 + 25129

Factor out common term

25

= 25 (- 1 + 129)

= \frac{25 ( - 1 + 129 )}{200}

Cancel the common factor:

25

= \frac{- 1 + 129}{8}

u = \frac{- 25 - 25 129}{2 \cdot 100} : - \frac{1 + 129}{8}

\frac{- 25 - 25 129}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{- 25 - 25 129}{200}

Factor

- 25 - 25129 : - 25 (1 + 129)

- 25 - 25129

Rewrite as

= - 25 \cdot 1 - 25129

Factor out common term

25

= - 25 (1 + 129)

= - \frac{25 ( 1 + 129 )}{200}

Cancel the common factor:

25

= - \frac{1 + 129}{8}

The solutions to the quadratic equation are:

u = \frac{- 1 + 129}{8}, u = - \frac{1 + 129}{8}

u = \frac{- 1 + 129}{8}, u = - \frac{1 + 129}{8}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

0.25 + u - \frac{2}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{- 1 + 129}{8}, u = - \frac{1 + 129}{8}

Substitute back

u = cot (x)

cot (x) = \frac{- 1 + 129}{8}, cot (x) = - \frac{1 + 129}{8}

cot (x) = \frac{- 1 + 129}{8}, cot (x) = - \frac{1 + 129}{8}

cot (x) = \frac{- 1 + 129}{8} : x = arccot (\frac{- 1 + 129}{8}) + πn

cot (x) = \frac{- 1 + 129}{8}

Apply trig inverse properties

cot (x) = \frac{- 1 + 129}{8}

General solutions for

cot (x) = \frac{- 1 + 129}{8}

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

x = arccot (\frac{- 1 + 129}{8}) + πn

x = arccot (\frac{- 1 + 129}{8}) + πn

cot (x) = - \frac{1 + 129}{8} : x = arccot (- \frac{1 + 129}{8}) + πn

cot (x) = - \frac{1 + 129}{8}

Apply trig inverse properties

cot (x) = - \frac{1 + 129}{8}

General solutions for

cot (x) = - \frac{1 + 129}{8}

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

x = arccot (- \frac{1 + 129}{8}) + πn

x = arccot (- \frac{1 + 129}{8}) + πn

Combine all the solutions

x = arccot (\frac{- 1 + 129}{8}) + πn, x = arccot (- \frac{1 + 129}{8}) + πn

Show solutions in decimal form

x = 0.65766 \dots + πn, x = 2.56707 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cot(x)-2tan(x)=-0.25 ?
The general solution for cot(x)-2tan(x)=-0.25 is x=0.65766…+pin,x=2.56707…+pin