Trigonometry Examples

$\cot (\frac{x}{3} + \frac{π}{5}) = - \tan (3 \cdot x - \frac{π}{7})$

Step 1

Add $\tan (3 \cdot x - \frac{π}{7})$ to both sides of the equation.

$\cot (\frac{x}{3} + \frac{π}{5}) + \tan (3 \cdot x - \frac{π}{7}) = 0$

Step 2

Set the argument in $\cot (\frac{x}{3} + \frac{π}{5})$ equal to $π n$ to find where the expression is undefined.

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

Step 3

Solve for $x$ .

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Step 3.1

Subtract $\frac{π}{5}$ from both sides of the equation.

$\frac{x}{3} = π n - \frac{π}{5}$

Step 3.2

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (π n - \frac{π}{5})$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (π n - \frac{π}{5})$

Step 3.3.1.1.2

Rewrite the expression.

$x = 3 (π n - \frac{π}{5})$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $3 (π n - \frac{π}{5})$ .

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Step 3.3.2.1.1

Apply the distributive property.

$x = 3 (π n) + 3 (- \frac{π}{5})$

Step 3.3.2.1.2

Multiply $3 (- \frac{π}{5})$ .

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Step 3.3.2.1.2.1

Multiply $- 1$ by $3$ .

$x = 3 π n - 3 \frac{π}{5}$

Step 3.3.2.1.2.2

Combine $- 3$ and $\frac{π}{5}$ .

$x = 3 π n + \frac{- 3 π}{5}$

Step 3.3.2.1.3

Move the negative in front of the fraction.

$x = 3 π n - \frac{3 π}{5}$

Step 4

Set the argument in $\tan (3 \cdot x - \frac{π}{7})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$3 \cdot x - \frac{π}{7} = \frac{π}{2} + π n$ , for any integer $n$

Step 5

Solve for $x$ .

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Step 5.1

Multiply $3$ by $x$ .

$3 x - \frac{π}{7} = \frac{π}{2} + π n$

Step 5.2

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

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Step 5.2.1

Add $\frac{π}{7}$ to both sides of the equation.

$3 x = \frac{π}{2} + π n + \frac{π}{7}$

Step 5.2.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$3 x = π n + \frac{π}{2} \cdot \frac{7}{7} + \frac{π}{7}$

Step 5.2.3

To write $\frac{π}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x = π n + \frac{π}{2} \cdot \frac{7}{7} + \frac{π}{7} \cdot \frac{2}{2}$

Step 5.2.4

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.4.1

Multiply $\frac{π}{2}$ by $\frac{7}{7}$ .

$3 x = π n + \frac{π \cdot 7}{2 \cdot 7} + \frac{π}{7} \cdot \frac{2}{2}$

Step 5.2.4.2

Multiply $2$ by $7$ .

$3 x = π n + \frac{π \cdot 7}{14} + \frac{π}{7} \cdot \frac{2}{2}$

Step 5.2.4.3

Multiply $\frac{π}{7}$ by $\frac{2}{2}$ .

$3 x = π n + \frac{π \cdot 7}{14} + \frac{π \cdot 2}{7 \cdot 2}$

Step 5.2.4.4

Multiply $7$ by $2$ .

$3 x = π n + \frac{π \cdot 7}{14} + \frac{π \cdot 2}{14}$

Step 5.2.5

Combine the numerators over the common denominator.

$3 x = π n + \frac{π \cdot 7 + π \cdot 2}{14}$

Step 5.2.6

Reorder $π$ and $2$ .

$3 x = π n + \frac{7 \cdot π + 2 \cdot π}{14}$

Step 5.2.7

Add $7 \cdot π$ and $2 \cdot π$ .

$3 x = π n + \frac{9 π}{14}$

Step 5.3

Divide each term in $3 x = π n + \frac{9 π}{14}$ by $3$ and simplify.

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Step 5.3.1

Divide each term in $3 x = π n + \frac{9 π}{14}$ by $3$ .

$\frac{3 x}{3} = \frac{π n}{3} + \frac{\frac{9 π}{14}}{3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{π n}{3} + \frac{\frac{9 π}{14}}{3}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{3} + \frac{\frac{9 π}{14}}{3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π n}{3} + \frac{9 π}{14} \cdot \frac{1}{3}$

Step 5.3.3.1.2

Cancel the common factor of $3$ .

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Step 5.3.3.1.2.1

Factor $3$ out of $9 π$ .

$x = \frac{π n}{3} + \frac{3 (3 π)}{14} \cdot \frac{1}{3}$

Step 5.3.3.1.2.2

Cancel the common factor.

$x = \frac{π n}{3} + \frac{3 (3 π)}{14} \cdot \frac{1}{3}$

Step 5.3.3.1.2.3

Rewrite the expression.

$x = \frac{π n}{3} + \frac{3 π}{14}$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = 3 π n - \frac{3 π}{5}, x = \frac{π n}{3} + \frac{3 π}{14}}$ $n$ , for any integer $n$

Step 7