Trigonometry Examples

\tan (θ) = 4

$\tan (θ) = 4$

Step 1

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

θ

$θ$ from inside the tangent.

θ = \arctan (4)

$θ = \arctan (4)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate

\arctan (4)

$\arctan (4)$ .

θ = 75.96375653

$θ = 75.96375653$

θ = 75.96375653

$θ = 75.96375653$

Step 3

The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from

180

$180$ to find the solution in the fourth quadrant.

θ = 180 + 75.96375653

$θ = 180 + 75.96375653$

Step 4

Add

180

$180$ and

75.96375653

$75.96375653$ .

θ = 255.96375653

$θ = 255.96375653$

Step 5

Find the period of

\tan (θ)

$\tan (θ)$ .

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Step 5.1

The period of the function can be calculated using

\frac{180}{| b |}

$\frac{180}{| b |}$ .

\frac{180}{| b |}

$\frac{180}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{180}{| 1 |}

$\frac{180}{| 1 |}$

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{180}{1}

$\frac{180}{1}$

Step 5.4

Divide

180

$180$ by

1

$1$ .

180

$180$

180

$180$

Step 6

The period of the

\tan (θ)

$\tan (θ)$ function is

180

$180$ so values will repeat every

180

$180$ degrees in both directions.

θ = 75.96375653 + 180 n, 255.96375653 + 180 n

$θ = 75.96375653 + 180 n, 255.96375653 + 180 n$ , for any integer

n

$n$

Step 7

Consolidate the answers.

θ = 75.96375653 + 180 n

$θ = 75.96375653 + 180 n$ , for any integer

n

$n$