Trigonometry Examples

$4 \csc^{2} (x) + 4 \csc (x) + 1 = 0$

Step 1

Substitute $u$ for $\csc (x)$ .

$4 {(u)}^{2} + 4 (u) + 1 = 0$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $4 u^{2}$ as ${(2 u)}^{2}$ .

${(2 u)}^{2} + 4 u + 1 = 0$

Step 2.2

Rewrite $1$ as $1^{2}$ .

${(2 u)}^{2} + 4 u + 1^{2} = 0$

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot (2 u) \cdot 1$

Step 2.4

Rewrite the polynomial.

${(2 u)}^{2} + 2 \cdot (2 u) \cdot 1 + 1^{2} = 0$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 u$ and $b = 1$ .

${(2 u + 1)}^{2} = 0$

Step 3

Set the $2 u + 1$ equal to $0$ .

$2 u + 1 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Step 4.2

Divide each term in $2 u = - 1$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 u = - 1$ by $2$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $u$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

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

Step 5

Substitute $\csc (x)$ for $u$ .

$\csc (x) = - \frac{1}{2}$

Step 6

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{1}{2}$ does not fall in this range, there is no solution.

No solution