Trigonometry Examples

$\csc (x + 53) = \sec (2 x + 25)$

Step 1

Subtract $\sec (2 x + 25)$ from both sides of the equation.

$\csc (x + 53) - \sec (2 x + 25) = 0$

Step 2

Set the argument in $\csc (x + 53)$ equal to $π n$ to find where the expression is undefined.

$x + 53 = π n$ , for any integer $n$

Step 3

Subtract $53$ from both sides of the equation.

$x = π n - 53$

Step 4

Set the argument in $\sec (2 x + 25)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$2 x + 25 = \frac{π}{2} + π n$ , for any integer $n$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $25$ from both sides of the equation.

$2 x = \frac{π}{2} + π n - 25$

Step 5.2

Divide each term in $2 x = \frac{π}{2} + π n - 25$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 x = \frac{π}{2} + π n - 25$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 5.2.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{π n}{2} + \frac{- 25}{2}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

$x = \frac{π}{4} + \frac{π n}{2} - \frac{25}{2}$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = π n - 53, x = \frac{π}{4} + \frac{π n}{2} - \frac{25}{2}}$ $n$ , for any integer $n$

Step 7