Trigonometry Examples

${(4 - \cos (10 x))}^{- 11}$

Step 1

Find the domain to determine if the expression is continuous.

Tap for more steps...

Step 1.1

Set the base in ${(4 - \cos (10 x))}^{- 11}$ equal to $0$ to find where the expression is undefined.

$4 - \cos (10 x) = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $4$ from both sides of the equation.

$- \cos (10 x) = - 4$

Step 1.2.2

Divide each term in $- \cos (10 x) = - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $- \cos (10 x) = - 4$ by $- 1$ .

$\frac{- \cos (10 x)}{- 1} = \frac{- 4}{- 1}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (10 x)}{1} = \frac{- 4}{- 1}$

Step 1.2.2.2.2

Divide $\cos (10 x)$ by $1$ .

$\cos (10 x) = \frac{- 4}{- 1}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $- 4$ by $- 1$ .

$\cos (10 x) = 4$

Step 1.2.3

The range of cosine is $- 1 \leq y \leq 1$ . Since $4$ does not fall in this range, there is no solution.

No solution

Step 1.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, ${(4 - \cos (10 x))}^{- 11}$ is continuous over all real numbers.

Continuous

Step 3