Precalculus Examples

$\tan (x) = \frac{1}{2}$ , $\sin (x) = x$

Step 1

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

$x = \arctan (\frac{1}{2})$

Step 2

Simplify the right side.

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Evaluate $\arctan (\frac{1}{2})$ .

$x = 0.4636476$

Step 3

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

$x = (3.14159265) + 0.4636476$

Step 4

Solve for $x$ .

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Remove parentheses.

$x = 3.14159265 + 0.4636476$

Remove parentheses.

$x = (3.14159265) + 0.4636476$

Add $3.14159265$ and $0.4636476$ .

$x = 3.60524026$

Step 5

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 6

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

$x = 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 7

Consolidate $0.4636476 + π n$ and $3.60524026 + π n$ to $0.4636476 + π n$ .

$x = 0.4636476 + π n$ , for any integer $n$

Step 8

Replace all occurrences of $x$ with $0.4636476 + π n$ in each equation.

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Replace all occurrences of $x$ in $\sin (x) = x$ with $0.4636476 + π n$ .

$\sin (0.4636476 + π n) = 0.4636476 + π n$

$x = 0.4636476 + π n$

Simplify the left side.

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Remove parentheses.

$\sin (0.4636476 + π n) = 0.4636476 + π n$

$x = 0.4636476 + π n$

$\sin (0.4636476 + π n) = 0.4636476 + π n$

$x = 0.4636476 + π n$

$\sin (0.4636476 + π n) = 0.4636476 + π n$

$x = 0.4636476 + π n$

Step 9

Solve the equation for $π$ .

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Subtract $\sin (0.4636476 + π n)$ from both sides of the equation.

$0 = 0.4636476 + π n - \sin (0.4636476 + π n)$

$x = 0.4636476 + π n$

Rewrite the equation as $0.4636476 + π n - \sin (0.4636476 + π n) = 0$ .

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

$x = 0.4636476 + π n$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

$x = 0.4636476 + π n$

Step 10

Solve the equation for $x$ .

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Multiply $0$ by $n$ .

$x = 0.4636476 + 0 \cdot n$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

Simplify $0.4636476 + 0 \cdot n$ .

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Multiply $0$ by $n$ .

$x = 0.4636476 + 0$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

Add $0.4636476$ and $0$ .

$x = 0.4636476$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

$x = 0.4636476$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

$x = 0.4636476$

$0.4636476 + π n - \sin (0.4636476 + π n) = 0$

Step 11

Solve the equation for $n$ .

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Simplify the left side.

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Simplify $\sin (0.4636476 + 0 \cdot n)$ .

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Rewrite.

$0 + 0 + \sin (0.4636476 + 0 \cdot n) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

Simplify by adding zeros.

$\sin (0.4636476 + 0 \cdot n) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

Multiply $0$ by $n$ .

$\sin (0.4636476 + 0) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

Add $0.4636476$ and $0$ .

$\sin (0.4636476) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

$\sin (0.4636476) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

$\sin (0.4636476) = 0.4636476 + 0 \cdot n$

$x = 0.4636476$

Simplify the right side.

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Simplify $0.4636476 + 0 \cdot n$ .

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Multiply $0$ by $n$ .

$\sin (0.4636476) = 0.4636476 + 0$

$x = 0.4636476$

Add $0.4636476$ and $0$ .

$\sin (0.4636476) = 0.4636476$

$x = 0.4636476$

$\sin (0.4636476) = 0.4636476$

$x = 0.4636476$

$\sin (0.4636476) = 0.4636476$

$x = 0.4636476$

Since $\sin (0.4636476) \neq 0.4636476$ , there are no solutions.

No solution

$x = 0.4636476$

No solution