What is the general solution for (1+cot(p))tan(p)=3 ?

The general solution for (1+cot(p))tan(p)=3 is p=1.10714…+pin

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Popular Trigonometry >

(1+cot(p))tan(p)=3

Solution

(1 + cot (p)) tan (p) = 3

Solution

p = 1.10714 \dots + πn

Degrees

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

Solution steps

(1 + cot (p)) tan (p) = 3

Subtract

3

from both sides

tan (p) (1 + cot (p)) - 3 = 0

Rewrite using trig identities

- 3 + (1 + cot (p)) tan (p)

Use the basic trigonometric identity:

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

= - 3 + (1 + cot (p)) \frac{1}{cot ( p )}

(1 + cot (p)) \frac{1}{cot ( p )} = \frac{1 + cot ( p )}{cot ( p )}

(1 + cot (p)) \frac{1}{cot ( p )}

Multiply fractions:

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

= \frac{1 \cdot ( 1 + cot ( p ) )}{cot ( p )}

1 \cdot (1 + cot (p)) = 1 + cot (p)

1 \cdot (1 + cot (p))

Multiply:

1 \cdot (1 + cot (p)) = (1 + cot (p))

= (1 + cot (p))

Remove parentheses:

(a) = a

= 1 + cot (p)

= \frac{1 + cot ( p )}{cot ( p )}

= - 3 + \frac{1 + cot ( p )}{cot ( p )}

- 3 + \frac{1 + cot ( p )}{cot ( p )} = 0

Solve by substitution

- 3 + \frac{1 + cot ( p )}{cot ( p )} = 0

Let:

cot (p) = u

- 3 + \frac{1 + u}{u} = 0

- 3 + \frac{1 + u}{u} = 0 : u = \frac{1}{2}

- 3 + \frac{1 + u}{u} = 0

Multiply both sides by

u

- 3 + \frac{1 + u}{u} = 0

Multiply both sides by

u

- 3 u + \frac{1 + u}{u} u = 0 \cdot u

Simplify

- 3 u + \frac{1 + u}{u} u = 0 \cdot u

Simplify

\frac{1 + u}{u} u : 1 + u

\frac{1 + u}{u} u

Multiply fractions:

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

= \frac{( 1 + u ) u}{u}

Cancel the common factor:

u

= 1 + u

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 3 u + 1 + u = 0

Simplify

- 3 u + 1 + u : - 2 u + 1

- 3 u + 1 + u

Group like terms

= - 3 u + u + 1

Add similar elements:

- 3 u + u = - 2 u

= - 2 u + 1

- 2 u + 1 = 0

- 2 u + 1 = 0

- 2 u + 1 = 0

Move

1

to the right side

- 2 u + 1 = 0

Subtract

1

from both sides

- 2 u + 1 - 1 = 0 - 1

Simplify

- 2 u = - 1

- 2 u = - 1

Divide both sides by

- 2

- 2 u = - 1

Divide both sides by

- 2

\frac{- 2 u}{- 2} = \frac{- 1}{- 2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 3 + \frac{1 + u}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}

Substitute back

u = cot (p)

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

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

cot (p) = \frac{1}{2} : p = arccot (\frac{1}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

p = arccot (\frac{1}{2}) + πn

p = arccot (\frac{1}{2}) + πn

Combine all the solutions

p = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

p = 1.10714 \dots + πn

Graph

Popular Examples

tan(t)= 7/24 ,0<t< pi/2

Frequently Asked Questions (FAQ)

What is the general solution for (1+cot(p))tan(p)=3 ?
The general solution for (1+cot(p))tan(p)=3 is p=1.10714…+pin