What is the general solution for 3tan(t)-cot(t)-5=0 ?

The general solution for 3tan(t)-cot(t)-5=0 is t=2.96305…+pin,t=1.07459…+pin

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3tan(t)-cot(t)-5=0

Solution

3 tan (t) - cot (t) - 5 = 0

Solution

t = 2.96305 \dots + πn, t = 1.07459 \dots + πn

Degrees

t = 169.77047 \dots^{\circ} + 18 0^{\circ} n, t = 61.56971 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 tan (t) - cot (t) - 5 = 0

Rewrite using trig identities

- 5 - cot (t) + 3 tan (t)

Use the basic trigonometric identity:

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

= - 5 - cot (t) + 3 \cdot \frac{1}{cot ( t )}

3 \cdot \frac{1}{cot ( t )} = \frac{3}{cot ( t )}

3 \cdot \frac{1}{cot ( t )}

Multiply fractions:

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

= \frac{1 \cdot 3}{cot ( t )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{cot ( t )}

= - 5 - cot (t) + \frac{3}{cot ( t )}

- 5 - cot (t) + \frac{3}{cot ( t )} = 0

Solve by substitution

- 5 - cot (t) + \frac{3}{cot ( t )} = 0

Let:

cot (t) = u

- 5 - u + \frac{3}{u} = 0

- 5 - u + \frac{3}{u} = 0 : u = - \frac{5 + 37}{2}, u = \frac{37 - 5}{2}

- 5 - u + \frac{3}{u} = 0

Multiply both sides by

u

- 5 - u + \frac{3}{u} = 0

Multiply both sides by

u

- 5 u - uu + \frac{3}{u} u = 0 \cdot u

Simplify

- 5 u - uu + \frac{3}{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{3}{u} u : 3

\frac{3}{u} u

Multiply fractions:

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

= \frac{3 u}{u}

Cancel the common factor:

u

= 3

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 5 u - u^{2} + 3 = 0

- 5 u - u^{2} + 3 = 0

- 5 u - u^{2} + 3 = 0

Solve

- 5 u - u^{2} + 3 = 0 : u = - \frac{5 + 37}{2}, u = \frac{37 - 5}{2}

- 5 u - u^{2} + 3 = 0

Write in the standard form

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

- u^{2} - 5 u + 3 = 0

Solve with the quadratic formula

- u^{2} - 5 u + 3 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 5, c = 3

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 1 ) \cdot 3}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 1 ) \cdot 3}{2 ( - 1 )}

(- 5)^{2} - 4 (- 1) \cdot 3 = 37

(- 5)^{2} - 4 (- 1) \cdot 3

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 1 \cdot 3

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 5^{2} + 12

5^{2} = 25

= 25 + 12

Add the numbers:

25 + 12 = 37

= 37

u_{1, 2} = \frac{- ( - 5 ) \pm 37}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 37}{2 ( - 1 )}, u_{2} = \frac{- ( - 5 ) - 37}{2 ( - 1 )}

u = \frac{- ( - 5 ) + 37}{2 ( - 1 )} : - \frac{5 + 37}{2}

\frac{- ( - 5 ) + 37}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 + 37}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{5 + 37}{- 2}

Apply the fraction rule:

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

= - \frac{5 + 37}{2}

u = \frac{- ( - 5 ) - 37}{2 ( - 1 )} : \frac{37 - 5}{2}

\frac{- ( - 5 ) - 37}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 - 37}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{5 - 37}{- 2}

Apply the fraction rule:

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

5 - 37 = - (37 - 5)

= \frac{37 - 5}{2}

The solutions to the quadratic equation are:

u = - \frac{5 + 37}{2}, u = \frac{37 - 5}{2}

u = - \frac{5 + 37}{2}, u = \frac{37 - 5}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 5 - u + \frac{3}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{5 + 37}{2}, u = \frac{37 - 5}{2}

Substitute back

u = cot (t)

cot (t) = - \frac{5 + 37}{2}, cot (t) = \frac{37 - 5}{2}

cot (t) = - \frac{5 + 37}{2}, cot (t) = \frac{37 - 5}{2}

cot (t) = - \frac{5 + 37}{2} : t = arccot (- \frac{5 + 37}{2}) + πn

cot (t) = - \frac{5 + 37}{2}

Apply trig inverse properties

cot (t) = - \frac{5 + 37}{2}

General solutions for

cot (t) = - \frac{5 + 37}{2}

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

t = arccot (- \frac{5 + 37}{2}) + πn

t = arccot (- \frac{5 + 37}{2}) + πn

cot (t) = \frac{37 - 5}{2} : t = arccot (\frac{37 - 5}{2}) + πn

cot (t) = \frac{37 - 5}{2}

Apply trig inverse properties

cot (t) = \frac{37 - 5}{2}

General solutions for

cot (t) = \frac{37 - 5}{2}

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

t = arccot (\frac{37 - 5}{2}) + πn

t = arccot (\frac{37 - 5}{2}) + πn

Combine all the solutions

t = arccot (- \frac{5 + 37}{2}) + πn, t = arccot (\frac{37 - 5}{2}) + πn

Show solutions in decimal form

t = 2.96305 \dots + πn, t = 1.07459 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan(t)-cot(t)-5=0 ?
The general solution for 3tan(t)-cot(t)-5=0 is t=2.96305…+pin,t=1.07459…+pin