Solve for θ in Degrees 25csc(theta)^2-4=0

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Trigonometry

$25 \csc^{2} (θ) - 4 = 0$

Add $4$ to both sides of the equation.

$25 \csc^{2} (θ) = 4$

Divide each term in $25 \csc^{2} (θ) = 4$ by $25$ and simplify.

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Divide each term in $25 \csc^{2} (θ) = 4$ by $25$ .

$\frac{25 \csc^{2} (θ)}{25} = \frac{4}{25}$

Simplify the left side.

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Cancel the common factor of $25$ .

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Cancel the common factor.

$\frac{25 \csc^{2} (θ)}{25} = \frac{4}{25}$

Divide $\csc^{2} (θ)$ by $1$ .

$\csc^{2} (θ) = \frac{4}{25}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\csc (θ) = \pm \sqrt{\frac{4}{25}}$

Simplify $\pm \sqrt{\frac{4}{25}}$ .

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Rewrite $\sqrt{\frac{4}{25}}$ as $\frac{\sqrt{4}}{\sqrt{25}}$ .

$\csc (θ) = \pm \frac{\sqrt{4}}{\sqrt{25}}$

Simplify the numerator.

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Rewrite $4$ as $2^{2}$ .

$\csc (θ) = \pm \frac{\sqrt{2^{2}}}{\sqrt{25}}$

Pull terms out from under the radical, assuming positive real numbers.

$\csc (θ) = \pm \frac{2}{\sqrt{25}}$

Simplify the denominator.

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Rewrite $25$ as $5^{2}$ .

$\csc (θ) = \pm \frac{2}{\sqrt{5^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\csc (θ) = \pm \frac{2}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$\csc (θ) = \frac{2}{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$\csc (θ) = - \frac{2}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

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

Set up each of the solutions to solve for $θ$ .

$\csc (θ) = \frac{2}{5}$

$\csc (θ) = - \frac{2}{5}$

Solve for $θ$ in $\csc (θ) = \frac{2}{5}$ .

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The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $\frac{2}{5}$ does not fall in this range, there is no solution.

No solution

Solve for $θ$ in $\csc (θ) = - \frac{2}{5}$ .

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The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{2}{5}$ does not fall in this range, there is no solution.

No solution

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Name

Name	two hundred twenty-three million seven hundred sixty-nine thousand three hundred twenty-three

Interesting facts

223769323 has 4 divisors, whose sum is 233498448
The reverse of 223769323 is 323967322
Previous prime number is 23

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 37
Digital Root 1

Name

Name	four hundred eighty-nine million eight hundred thirty-eight thousand six hundred forty-eight

Interesting facts

489838648 has 32 divisors, whose sum is 1780375464
The reverse of 489838648 is 846838984
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 58
Digital Root 4

Name

Name	one billion seven hundred sixty-four million two hundred thirty-six thousand two hundred fifty-six

Interesting facts

1764236256 has 128 divisors, whose sum is 17862893064
The reverse of 1764236256 is 6526324671
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 42
Digital Root 6

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