Trigonometry Examples

$\tan (- x) = \frac{1}{4}$ , $\sec (x)$

Step 1

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

$- x = \arctan (\frac{1}{4})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arctan (\frac{1}{4})$ .

$- x = 0.24497866$

Step 3

Divide each term in $- x = 0.24497866$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- x = 0.24497866$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0.24497866}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0.24497866}{- 1}$

Step 3.2.2

Divide $x$ by $1$ .

$x = \frac{0.24497866}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0.24497866$ by $- 1$ .

$x = - 0.24497866$

Step 4

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.

$- x = (3.14159265) + 0.24497866$

Step 5

Divide each term in $- x = (3.14159265) + 0.24497866$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x = (3.14159265) + 0.24497866$ by $- 1$ .

$\frac{- x}{- 1} = \frac{3.14159265}{- 1} + \frac{0.24497866}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{3.14159265}{- 1} + \frac{0.24497866}{- 1}$

Step 5.2.2

Divide $x$ by $1$ .

$x = \frac{3.14159265}{- 1} + \frac{0.24497866}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$x = \frac{3.14159265 + 0.24497866}{- 1}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Add $3.14159265$ and $0.24497866$ .

$x = \frac{3.38657131}{- 1}$

Step 5.3.2.2

Divide $3.38657131$ by $- 1$ .

$x = - 3.38657131$

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 $- 1$ and $0$ 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 $- 0.24497866$ to find the positive angle.

$- 0.24497866 + π$

Step 7.2

Replace with decimal approximation.

$3.14159265 - 0.24497866$

Step 7.3

Subtract $0.24497866$ from $3.14159265$ .

$2.89661399$

Step 7.4

Add $π$ to $- 3.38657131$ to find the positive angle.

$- 3.38657131 + π$

Step 7.5

Replace with decimal approximation.

$3.14159265 - 3.38657131$

Step 7.6

Subtract $3.38657131$ from $3.14159265$ .

$- 0.24497866$

Step 7.7

List the new angles.

$x = - 0.24497866$

Step 8

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

$x = - 0.24497866 + π n, 2.89661399 + π n$ , for any integer $n$

Step 9

Take the base solution.

$x = - 0.24497866$

Step 10

Replace the variable $x$ with $- 0.24497866$ in the expression.

$\sec (- 0.24497866)$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sec (- 0.24497866)$

Decimal Form:

$1.03077640 \dots$