Precalculus Examples

$\tan (x) = - \frac{12}{15}$ , $\tan (2 x) = y$

Step 1

Cancel the common factor of $12$ and $15$ .

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Step 1.1

Factor $3$ out of $12$ .

$\tan (x) = - \frac{3 (4)}{15}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $15$ .

$\tan (x) = - \frac{3 \cdot 4}{3 \cdot 5}$

Step 1.2.2

Cancel the common factor.

$\tan (x) = - \frac{3 \cdot 4}{3 \cdot 5}$

Step 1.2.3

Rewrite the expression.

$\tan (x) = - \frac{4}{5}$

Step 2

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

$x = \arctan (- \frac{4}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (- \frac{4}{5})$ .

$x = - 0.67474094$

Step 4

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - 0.67474094 - (3.14159265)$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Add $2 π$ to $- 0.67474094 - (3.14159265)$ .

$x = - 0.67474094 - (3.14159265) + 2 π$

Step 5.2

The resulting angle of $2.46685171$ is positive and coterminal with $- 0.67474094 - (3.14159265)$ .

$x = 2.46685171$

Step 6

Find the period of $\tan (x)$ .

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 6.4

Divide $π$ by $1$ .

$π$

Step 7

Add $π$ to every negative angle to get positive angles.

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Step 7.1

Add $π$ to $- 0.67474094$ to find the positive angle.

$- 0.67474094 + π$

Step 7.2

Replace with decimal approximation.

$3.14159265 - 0.67474094$

Step 7.3

Subtract $0.67474094$ from $3.14159265$ .

$2.46685171$

Step 7.4

List the new angles.

$x = 2.46685171$

Step 8

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

$x = 2.46685171 + π n, 2.46685171 + π n$ , for any integer $n$

Step 9

Consolidate $2.46685171 + π n$ and $2.46685171 + π n$ to $2.46685171 + π n$ .

$x = 2.46685171 + π n$ , for any integer $n$

Step 10

Replace all occurrences of $x$ with $2.46685171 + π n$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $\tan (2 x) = y$ with $2.46685171 + π n$ .

$\tan (2 (2.46685171 + π n)) = y$

$x = 2.46685171 + π n$

Step 10.2

Simplify the left side.

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Step 10.2.1

Simplify $\tan (2 (2.46685171 + π n))$ .

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Step 10.2.1.1

Apply the distributive property.

$\tan (2 \cdot 2.46685171 + 2 (π n)) = y$

$x = 2.46685171 + π n$

Step 10.2.1.2

Multiply $2$ by $2.46685171$ .

$\tan (4.93370342 + 2 π n) = y$

$x = 2.46685171 + π n$

$\tan (4.93370342 + 2 π n) = y$

$x = 2.46685171 + π n$

$\tan (4.93370342 + 2 π n) = y$

$x = 2.46685171 + π n$

$\tan (4.93370342 + 2 π n) = y$

$x = 2.46685171 + π n$

Step 11

Solve the equation for $π$ .

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Step 11.1

Take the inverse tangent of both sides of the equation to extract $π$ from inside the tangent.

$4.93370342 + 2 π n = \arctan (y)$

$x = 2.46685171 + π n$

Step 11.2

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

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Step 11.2.1

Subtract $4.93370342$ from both sides of the equation.

$2 π n = \arctan (y) - 4.93370342$

$x = 2.46685171 + π n$

Step 11.2.2

Subtract $2 π n$ from both sides of the equation.

$0 = \arctan (y) - 4.93370342 - 2 π n$

$x = 2.46685171 + π n$

$0 = \arctan (y) - 4.93370342 - 2 π n$

$x = 2.46685171 + π n$

Step 11.3

Rewrite the equation as $\arctan (y) - 4.93370342 - 2 π n = 0$ .

$\arctan (y) - 4.93370342 - 2 π n = 0$

$x = 2.46685171 + π n$

Step 11.4

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

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Step 11.4.1

Subtract $4.93370342$ from both sides of the equation.

$2 π n = \arctan (y) - 4.93370342$

$x = 2.46685171 + π n$

Step 11.4.2

Subtract $2 π n$ from both sides of the equation.

$0 = \arctan (y) - 4.93370342 - 2 π n$

$x = 2.46685171 + π n$

$0 = \arctan (y) - 4.93370342 - 2 π n$

$x = 2.46685171 + π n$

Step 11.5

Rewrite the equation as $\arctan (y) - 4.93370342 - 2 π n = 0$ .

$\arctan (y) - 4.93370342 - 2 π n = 0$

$x = 2.46685171 + π n$

$\arctan (y) - 4.93370342 - 2 π n = 0$

$x = 2.46685171 + π n$

Step 12

Solve the equation for $n$ .

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Step 12.1

Rewrite the equation as $2.46685171 + 0 \cdot n = x$ .

$2.46685171 + 0 \cdot n = x$

$\arctan (y) - 4.93370342 - 2 π n = 0$

Step 12.2

Simplify $2.46685171 + 0 \cdot n$ .

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Step 12.2.1

Multiply $0$ by $n$ .

$2.46685171 + 0 = x$

$\arctan (y) - 4.93370342 - 2 π n = 0$

Step 12.2.2

Add $2.46685171$ and $0$ .

$2.46685171 = x$

$\arctan (y) - 4.93370342 - 2 π n = 0$

$2.46685171 = x$

$\arctan (y) - 4.93370342 - 2 π n = 0$

Step 12.3

Rewrite $2.46685171 = x$ so $x$ is on the left side.

$x = 2.46685171$

$\arctan (y) - 4.93370342 - 2 π n = 0$

Step 12.4

The variable $n$ got canceled.

All real numbers

$\arctan (y) - 4.93370342 - 2 π n = 0$

All real numbers

$\arctan (y) - 4.93370342 - 2 π n = 0$

Step 13

Solve the equation for $y$ .

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Step 13.1

Rewrite the equation as $y = \tan (4.93370342 + 2 (0) \cdot (t a n l \cdot l (r e a l) (n u m b e r s)))$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a n l \cdot l (r e a l) (n u m b e r s)))$

Step 13.2

Simplify $\tan (4.93370342 + 2 (0) \cdot (t a n l \cdot l (r e a l) (n u m b e r s)))$ .

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 13.2.1.1.1

Move $a$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t (a \cdot a) n l \cdot l (r e l) (n u m b e r s)))$

Step 13.2.1.1.2

Multiply $a$ by $a$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n l \cdot l (r e l) (n u m b e r s)))$

Step 13.2.1.2

Multiply $n$ by $n$ by adding the exponents.

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Step 13.2.1.2.1

Move $n$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} (n \cdot n) l \cdot l (r e l) (u m b e r s)))$

Step 13.2.1.2.2

Multiply $n$ by $n$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l \cdot l (r e l) (u m b e r s)))$

Step 13.2.1.3

Multiply $l$ by $l$ by adding the exponents.

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Step 13.2.1.3.1

Move $l$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} (l \cdot l) (r e l) (u m b e r s)))$

Step 13.2.1.3.2

Multiply $l$ by $l$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l^{2} (r e l) (u m b e r s)))$

Step 13.2.1.4

Multiply $l^{2}$ by $l$ by adding the exponents.

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Step 13.2.1.4.1

Move $l$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} (l \cdot l^{2}) (r e) (u m b e r s)))$

Step 13.2.1.4.2

Multiply $l$ by $l^{2}$ .

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Step 13.2.1.4.2.1

Raise $l$ to the power of $1$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} (l^{1} l^{2}) (r e) (u m b e r s)))$

Step 13.2.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l^{1 + 2} (r e) (u m b e r s)))$

Step 13.2.1.4.3

Add $1$ and $2$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l^{3} (r e) (u m b e r s)))$

Step 13.2.1.5

Multiply $r$ by $r$ by adding the exponents.

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Step 13.2.1.5.1

Move $r$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l^{3} (r \cdot r e) (u m b e s)))$

Step 13.2.1.5.2

Multiply $r$ by $r$ .

$y = \tan (4.93370342 + 2 (0) \cdot (t a^{2} n^{2} l^{3} (r^{2} e) (u m b e s)))$

Step 13.2.1.6

Multiply $2$ by $0$ .

$y = \tan (4.93370342 + 0 \cdot (t a^{2} n^{2} l^{3} (r^{2} e) (u m b e s)))$

Step 13.2.1.7

Rewrite using the commutative property of multiplication.

$y = \tan (4.93370342 + 0 \cdot (e t a^{2} n^{2} l^{3} r^{2} (u m b e s)))$

Step 13.2.1.8

Multiply $e t a^{2} n^{2} l^{3} r^{2} (u m b e s)$ .

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Step 13.2.1.8.1

Raise $e$ to the power of $1$ .

$y = \tan (4.93370342 + 0 \cdot (t a^{2} n^{2} l^{3} r^{2} (u m b (e^{1} e) s)))$

Step 13.2.1.8.2

Raise $e$ to the power of $1$ .

$y = \tan (4.93370342 + 0 \cdot (t a^{2} n^{2} l^{3} r^{2} (u m b (e^{1} e^{1}) s)))$

Step 13.2.1.8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \tan (4.93370342 + 0 \cdot (t a^{2} n^{2} l^{3} r^{2} (u m b e^{1 + 1} s)))$

Step 13.2.1.8.4

Add $1$ and $1$ .

$y = \tan (4.93370342 + 0 \cdot (t a^{2} n^{2} l^{3} r^{2} (u m b e^{2} s)))$

Step 13.2.1.9

Rewrite using the commutative property of multiplication.

$y = \tan (4.93370342 + 0 \cdot (e^{2} t a^{2} n^{2} l^{3} r^{2} u m b s))$

Step 13.2.1.10

Multiply $0 (e^{2} t a^{2} n^{2} l^{3} r^{2} u m b s)$ .

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Step 13.2.1.10.1

Multiply $e^{2}$ by $0$ .

$y = \tan (4.93370342 + 0 (t a^{2} n^{2} l^{3} r^{2} u m b s))$

Step 13.2.1.10.2

Multiply $t$ by $0$ .

$y = \tan (4.93370342 + 0 (a^{2} n^{2} l^{3} r^{2} u m b s))$

Step 13.2.1.10.3

Multiply $a^{2}$ by $0$ .

$y = \tan (4.93370342 + 0 (n^{2} l^{3} r^{2} u m b s))$

Step 13.2.1.10.4

Multiply $n^{2}$ by $0$ .

$y = \tan (4.93370342 + 0 (l^{3} r^{2} u m b s))$

Step 13.2.1.10.5

Multiply $l^{3}$ by $0$ .

$y = \tan (4.93370342 + 0 (r^{2} u m b s))$

Step 13.2.1.10.6

Multiply $r^{2}$ by $0$ .

$y = \tan (4.93370342 + 0 (u m b s))$

Step 13.2.1.10.7

Multiply $u$ by $0$ .

$y = \tan (4.93370342 + 0 (m b s))$

Step 13.2.1.10.8

Multiply $m$ by $0$ .

$y = \tan (4.93370342 + 0 (b s))$

Step 13.2.1.10.9

Multiply $b$ by $0$ .

$y = \tan (4.93370342 + 0 s)$

Step 13.2.1.10.10

Multiply $0$ by $s$ .

$y = \tan (4.93370342 + 0)$

Step 13.2.2

Add $4.93370342$ and $0$ .

$y = \tan (4.93370342)$