Trigonometry Examples

$\tan (x - \frac{π}{4}) = - \frac{1}{7}$

Step 1

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

$x - \frac{π}{4} = \arctan (- \frac{1}{7})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arctan (- \frac{1}{7})$ .

$x - \frac{π}{4} = - 0.14189705$

Step 3

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

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Step 3.1

Add $\frac{π}{4}$ to both sides of the equation.

$x = - 0.14189705 + \frac{π}{4}$

Step 3.2

Add $- 0.14189705$ and $\frac{π}{4}$ .

$x = 0.6435011$

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.

$x - \frac{π}{4} = - 0.14189705 - (3.14159265)$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Add $2 π$ to $- 0.14189705 - (3.14159265)$ .

$x - \frac{π}{4} = - 0.14189705 - (3.14159265) + 2 π$

Step 5.2

The resulting angle of $2.99969559$ is positive and coterminal with $- 0.14189705 - (3.14159265)$ .

$x - \frac{π}{4} = 2.99969559$

Step 5.3

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

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Step 5.3.1

Add $\frac{π}{4}$ to both sides of the equation.

$x = 2.99969559 + \frac{π}{4}$

Step 5.3.2

Add $2.99969559$ and $\frac{π}{4}$ .

$x = 3.78509376$

Step 6

Find the period of $\tan (x - \frac{π}{4})$ .

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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

The period of the $\tan (x - \frac{π}{4})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 0.6435011 + π n, 3.78509376 + π n$ , for any integer $n$

Step 8

Consolidate $0.6435011 + π n$ and $3.78509376 + π n$ to $0.6435011 + π n$ .

$x = 0.6435011 + π n$ , for any integer $n$