Precalculus Examples

$\sec (3 x) \cos (2 x) = 0$

Step 1

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

$\sec (3 x) = 0$

$\cos (2 x) = 0$

Step 2

Set $\sec (3 x)$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\sec (3 x)$ equal to $0$ .

$\sec (3 x) = 0$

Step 2.2

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

No solution

Step 3

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

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Step 3.1

Set $\cos (2 x)$ equal to $0$ .

$\cos (2 x) = 0$

Step 3.2

Solve $\cos (2 x) = 0$ for $x$ .

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Step 3.2.1

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

$2 x = \arccos (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$2 x = \frac{π}{2}$

Step 3.2.3

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

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Step 3.2.3.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 3.2.3.3.2

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

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Step 3.2.3.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 3.2.3.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 3.2.4

The cosine 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.

$2 x = 2 π - \frac{π}{2}$

Step 3.2.5

Solve for $x$ .

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Step 3.2.5.1

Simplify.

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Step 3.2.5.1.1

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

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

Step 3.2.5.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 3.2.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 3.2.5.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 3.2.5.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 3.2.5.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 3.2.5.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

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

Step 3.2.5.2.2

Simplify the left side.

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Step 3.2.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.5.2.2.1.1

Cancel the common factor.

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

Step 3.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 3.2.5.2.3

Simplify the right side.

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Step 3.2.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 3.2.5.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 3.2.5.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 3.2.5.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 3.2.6

Find the period of $\cos (2 x)$ .

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Step 3.2.6.1

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

$\frac{2 π}{| b |}$

Step 3.2.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.6.4.2

Divide $π$ by $1$ .

$π$

Step 3.2.7

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

$x = \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 4

The final solution is all the values that make $\sec (3 x) \cos (2 x) = 0$ true.

$x = \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 5

Consolidate the answers.

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