What is the general solution for csc(x)=-sqrt(1+cot(x)) ?

The general solution for csc(x)=-sqrt(1+cot(x)) is x= pi/4+pin,x= pi/2+pin

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Popular Trigonometry >

csc(x)=-sqrt(1+cot(x))

Solution

csc (x) = - 1 + cot (x)

Solution

x = \frac{π}{4} + πn, x = \frac{π}{2} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

csc (x) = - 1 + cot (x)

Square both sides

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

Subtract

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

from both sides

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

Rewrite using trig identities

- 1 - cot (x) + csc^{2} (x)

Use the Pythagorean identity:

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

csc^{2} (x) - 1 = cot^{2} (x)

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

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

Solve by substitution

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

Let:

cot (x) = u

- u + u^{2} = 0

- u + u^{2} = 0 : u = 1, u = 0

- u + u^{2} = 0

Write in the standard form

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

u^{2} - u = 0

Solve with the quadratic formula

u^{2} - u = 0

Quadratic Equation Formula:

For

a = 1, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 1)^{2} - 4 \cdot 1 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 1 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

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

\frac{- ( - 1 ) + 1}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{1 + 1}{2 \cdot 1}

Add the numbers:

1 + 1 = 2

= \frac{2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 1 ) - 1}{2 \cdot 1} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 1}

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 1, u = 0

Substitute back

u = cot (x)

cot (x) = 1, cot (x) = 0

cot (x) = 1, cot (x) = 0

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

General solutions for

cot (x) = 1

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

General solutions for

cot (x) = 0

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{π}{2} + πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

csc (x) = - 1 + cot (x)

Remove the ones that don't agree with the equation.

Check the solution

\frac{π}{4} + πn :

True

\frac{π}{4} + πn

Plug in

n = 1

\frac{π}{4} + π 1

For

csc (x) = - 1 + cot (x)

plug in

x = \frac{π}{4} + π 1

csc (\frac{π}{4} + π 1) = - 1 + cot (\frac{π}{4} + π 1)

Refine

- 1.41421 \dots = - 1.41421 \dots

\Rightarrow True

Check the solution

\frac{π}{2} + πn :

True

\frac{π}{2} + πn

Plug in

n = 1

\frac{π}{2} + π 1

For

csc (x) = - 1 + cot (x)

plug in

x = \frac{π}{2} + π 1

csc (\frac{π}{2} + π 1) = - 1 + cot (\frac{π}{2} + π 1)

Refine

- 1 = - 1

\Rightarrow True

x = \frac{π}{4} + πn, x = \frac{π}{2} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)=-sqrt(1+cot(x)) ?
The general solution for csc(x)=-sqrt(1+cot(x)) is x= pi/4+pin,x= pi/2+pin