Trigonometry Examples

$\cos (\arctan (1) + \arccos (\frac{x}{2}))$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the argument in $\arccos (\frac{x}{2})$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{x}{2} \geq - 1$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq - 1 \cdot 2$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq - 1 \cdot 2$

Step 1.2.2.1.1.2

Rewrite the expression.

$x \geq - 1 \cdot 2$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $- 1$ by $2$ .

$x \geq - 2$

Step 1.3

Set the argument in $\arccos (\frac{x}{2})$ less than or equal to $1$ to find where the expression is defined.

$\frac{x}{2} \leq 1$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \leq 1 \cdot 2$

Step 1.4.2

Simplify.

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Step 1.4.2.1

Simplify the left side.

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Step 1.4.2.1.1

Cancel the common factor of $2$ .

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Step 1.4.2.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \leq 1 \cdot 2$

Step 1.4.2.1.1.2

Rewrite the expression.

$x \leq 1 \cdot 2$

Step 1.4.2.2

Simplify the right side.

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Step 1.4.2.2.1

Multiply $2$ by $1$ .

$x \leq 2$

Step 1.5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Step 2

The expression is continuous.

Continuous

Step 3