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Popular >

3csc^2(x)+cot^2(x)-4=0

Solution

3 csc^{2} (x) + cot^{2} (x) - 4 = 0

Solution

x = 1.10714 \dots + 2 πn, x = π - 1.10714 \dots + 2 πn, x = - 1.10714 \dots + 2 πn, x = π + 1.10714 \dots + 2 πn

Degrees

x = 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 243.43494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 csc^{2} (x) + cot^{2} (x) - 4 = 0

Rewrite using trig identities

- 4 + cot^{2} (x) + 3 csc^{2} (x)

Use the Pythagorean identity:

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

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

= - 4 + csc^{2} (x) - 1 + 3 csc^{2} (x)

Simplify

- 4 + csc^{2} (x) - 1 + 3 csc^{2} (x) : 4 csc^{2} (x) - 5

- 4 + csc^{2} (x) - 1 + 3 csc^{2} (x)

Group like terms

= csc^{2} (x) + 3 csc^{2} (x) - 4 - 1

Add similar elements:

csc^{2} (x) + 3 csc^{2} (x) = 4 csc^{2} (x)

= 4 csc^{2} (x) - 4 - 1

Subtract the numbers:

- 4 - 1 = - 5

= 4 csc^{2} (x) - 5

= 4 csc^{2} (x) - 5

- 5 + 4 csc^{2} (x) = 0

Solve by substitution

- 5 + 4 csc^{2} (x) = 0

Let:

csc (x) = u

- 5 + 4 u^{2} = 0

- 5 + 4 u^{2} = 0 : u = \frac{5}{2}, u = - \frac{5}{2}

- 5 + 4 u^{2} = 0

Move

5

to the right side

- 5 + 4 u^{2} = 0

Add

5

to both sides

- 5 + 4 u^{2} + 5 = 0 + 5

Simplify

4 u^{2} = 5

4 u^{2} = 5

Divide both sides by

4

4 u^{2} = 5

Divide both sides by

4

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{5}{4}, u = - \frac{5}{4}

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

\frac{5}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{5}{2}

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

- \frac{5}{4}

Simplify

\frac{5}{4} : \frac{5}{2}

\frac{5}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{5}{2}

= - \frac{5}{2}

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

Substitute back

u = csc (x)

csc (x) = \frac{5}{2}, csc (x) = - \frac{5}{2}

csc (x) = \frac{5}{2}, csc (x) = - \frac{5}{2}

csc (x) = \frac{5}{2} : x = arccsc (\frac{5}{2}) + 2 πn, x = π - arccsc (\frac{5}{2}) + 2 πn

csc (x) = \frac{5}{2}

Apply trig inverse properties

csc (x) = \frac{5}{2}

General solutions for

csc (x) = \frac{5}{2}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (\frac{5}{2}) + 2 πn, x = π - arccsc (\frac{5}{2}) + 2 πn

x = arccsc (\frac{5}{2}) + 2 πn, x = π - arccsc (\frac{5}{2}) + 2 πn

csc (x) = - \frac{5}{2} : x = arccsc (- \frac{5}{2}) + 2 πn, x = π + arccsc (\frac{5}{2}) + 2 πn

csc (x) = - \frac{5}{2}

Apply trig inverse properties

csc (x) = - \frac{5}{2}

General solutions for

csc (x) = - \frac{5}{2}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- \frac{5}{2}) + 2 πn, x = π + arccsc (\frac{5}{2}) + 2 πn

x = arccsc (- \frac{5}{2}) + 2 πn, x = π + arccsc (\frac{5}{2}) + 2 πn

Combine all the solutions

x = arccsc (\frac{5}{2}) + 2 πn, x = π - arccsc (\frac{5}{2}) + 2 πn, x = arccsc (- \frac{5}{2}) + 2 πn, x = π + arccsc (\frac{5}{2}) + 2 πn

Show solutions in decimal form

x = 1.10714 \dots + 2 πn, x = π - 1.10714 \dots + 2 πn, x = - 1.10714 \dots + 2 πn, x = π + 1.10714 \dots + 2 πn

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