Trigonometry Examples

$arcsec (\frac{x}{3} - \frac{1}{2}) = \frac{π}{3}$

Step 1

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

$\frac{x}{3} - \frac{1}{2} = \sec (\frac{π}{3})$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{3} - \frac{1}{2} = 2$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $\frac{1}{2}$ to both sides of the equation.

$\frac{x}{3} = 2 + \frac{1}{2}$

Step 3.2

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

$\frac{x}{3} = 2 \cdot \frac{2}{2} + \frac{1}{2}$

Step 3.3

Combine $2$ and $\frac{2}{2}$ .

$\frac{x}{3} = \frac{2 \cdot 2}{2} + \frac{1}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x}{3} = \frac{2 \cdot 2 + 1}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

$\frac{x}{3} = \frac{4 + 1}{2}$

Step 3.5.2

Add $4$ and $1$ .

$\frac{x}{3} = \frac{5}{2}$

Step 4

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (\frac{5}{2})$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (\frac{5}{2})$

Step 5.1.1.2

Rewrite the expression.

$x = 3 (\frac{5}{2})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $3 (\frac{5}{2})$ .

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Step 5.2.1.1

Combine $3$ and $\frac{5}{2}$ .

$x = \frac{3 \cdot 5}{2}$

Step 5.2.1.2

Multiply $3$ by $5$ .

$x = \frac{15}{2}$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Add $\frac{1}{2}$ to both sides of the equation.

$\frac{x}{3} = 2 + \frac{1}{2}$

Step 6.2

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

$\frac{x}{3} = 2 \cdot \frac{2}{2} + \frac{1}{2}$

Step 6.3

Combine $2$ and $\frac{2}{2}$ .

$\frac{x}{3} = \frac{2 \cdot 2}{2} + \frac{1}{2}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{x}{3} = \frac{2 \cdot 2 + 1}{2}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $2$ by $2$ .

$\frac{x}{3} = \frac{4 + 1}{2}$

Step 6.5.2

Add $4$ and $1$ .

$\frac{x}{3} = \frac{5}{2}$

Step 7

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (\frac{5}{2})$

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Cancel the common factor of $3$ .

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Step 8.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (\frac{5}{2})$

Step 8.1.1.2

Rewrite the expression.

$x = 3 (\frac{5}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Multiply $3 (\frac{5}{2})$ .

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Step 8.2.1.1

Combine $3$ and $\frac{5}{2}$ .

$x = \frac{3 \cdot 5}{2}$

Step 8.2.1.2

Multiply $3$ by $5$ .

$x = \frac{15}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{2}$

Decimal Form:

$x = 7.5$

Mixed Number Form:

$x = 7 \frac{1}{2}$