Trigonometry Examples

$\sec (x) \csc (x) = 2 \csc (x)$

Step 1

Subtract $2 \csc (x)$ from both sides of the equation.

$\sec (x) \csc (x) - 2 \csc (x) = 0$

Step 2

Factor $\csc (x)$ out of $\sec (x) \csc (x) - 2 \csc (x)$ .

Tap for more steps...

Step 2.1

Factor $\csc (x)$ out of $\sec (x) \csc (x)$ .

$\csc (x) \sec (x) - 2 \csc (x) = 0$

Step 2.2

Factor $\csc (x)$ out of $- 2 \csc (x)$ .

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

Step 2.3

Factor $\csc (x)$ out of $\csc (x) \sec (x) + \csc (x) \cdot - 2$ .

$\csc (x) (\sec (x) - 2) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\csc (x) = 0$

$\sec (x) - 2 = 0$

Step 4

Set $\csc (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $\csc (x)$ equal to $0$ .

$\csc (x) = 0$

Step 4.2

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 5

Set $\sec (x) - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $\sec (x) - 2$ equal to $0$ .

$\sec (x) - 2 = 0$

Step 5.2

Solve $\sec (x) - 2 = 0$ for $x$ .

Tap for more steps...

Step 5.2.1

Add $2$ to both sides of the equation.

$\sec (x) = 2$

Step 5.2.2

Take the inverse secant of both sides of the equation to extract $x$ from inside the secant.

$x = arcsec (2)$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 5.2.4

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{3}$

Step 5.2.5

Simplify $2 π - \frac{π}{3}$ .

Tap for more steps...

Step 5.2.5.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 5.2.5.2

Combine fractions.

Tap for more steps...

Step 5.2.5.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 5.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 5.2.5.3

Simplify the numerator.

Tap for more steps...

Step 5.2.5.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 5.2.5.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 5.2.6

Find the period of $\sec (x)$ .

Tap for more steps...

Step 5.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $\csc (x) (\sec (x) - 2) = 0$ true.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$