Trigonometry Examples

\tan (x) = - \frac{2}{7}

$\tan (x) = - \frac{2}{7}$

Step 1

Take the inverse tangent of both sides of the equation to extract

x

$x$ from inside the tangent.

x = \arctan (- \frac{2}{7})

$x = \arctan (- \frac{2}{7})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate

\arctan (- \frac{2}{7})

$\arctan (- \frac{2}{7})$ .

x = - 0.27829965

$x = - 0.27829965$

x = - 0.27829965

$x = - 0.27829965$

Step 3

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

π

$π$ to find the solution in the third quadrant.

x = - 0.27829965 - (3.14159265)

$x = - 0.27829965 - (3.14159265)$

Step 4

Simplify the expression to find the second solution.

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Step 4.1

Add

2 π

$2 π$ to

- 0.27829965 - (3.14159265)

$- 0.27829965 - (3.14159265)$ .

x = - 0.27829965 - (3.14159265) + 2 π

$x = - 0.27829965 - (3.14159265) + 2 π$

Step 4.2

The resulting angle of

2.86329299

$2.86329299$ is positive and coterminal with

- 0.27829965 - (3.14159265)

$- 0.27829965 - (3.14159265)$ .

x = 2.86329299

$x = 2.86329299$

x = 2.86329299

$x = 2.86329299$

Step 5

Find the period of

\tan (x)

$\tan (x)$ .

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Step 5.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{π}{| 1 |}

$\frac{π}{| 1 |}$

Step 5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{π}{1}

$\frac{π}{1}$

Step 5.4

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

Step 6

Add

π

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

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Step 6.1

Add

π

$π$ to

- 0.27829965

$- 0.27829965$ to find the positive angle.

- 0.27829965 + π

$- 0.27829965 + π$

Step 6.2

Replace with decimal approximation.

3.14159265 - 0.27829965

$3.14159265 - 0.27829965$

Step 6.3

Subtract

0.27829965

$0.27829965$ from

3.14159265

$3.14159265$ .

2.86329299

$2.86329299$

Step 6.4

List the new angles.

x = 2.86329299

$x = 2.86329299$

x = 2.86329299

$x = 2.86329299$

Step 7

The period of the

\tan (x)

$\tan (x)$ function is

π

$π$ so values will repeat every

π

$π$ radians in both directions.

x = 2.86329299 + π n, 2.86329299 + π n

$x = 2.86329299 + π n, 2.86329299 + π n$ , for any integer

n

$n$

Step 8

Consolidate

2.86329299 + π n

$2.86329299 + π n$ and

2.86329299 + π n

$2.86329299 + π n$ to

2.86329299 + π n

$2.86329299 + π n$ .

x = 2.86329299 + π n

$x = 2.86329299 + π n$ , for any integer

n

$n$