Trigonometry Examples

$θ = \arctan (\frac{2 t}{t^{2}})$

Step 1

Rewrite the equation as $\arctan (\frac{2 t}{t^{2}}) = θ$ .

$\arctan (\frac{2 t}{t^{2}}) = θ$

Step 2

Take the inverse arctangent of both sides of the equation to extract $t$ from inside the arctangent.

$\frac{2 t}{t^{2}} = \tan (θ)$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $t$ and $t^{2}$ .

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Step 3.1.1

Factor $t$ out of $2 t$ .

$\frac{t \cdot 2}{t^{2}} = \tan (θ)$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $t$ out of $t^{2}$ .

$\frac{t \cdot 2}{t \cdot t} = \tan (θ)$

Step 3.1.2.2

Cancel the common factor.

$\frac{t \cdot 2}{t \cdot t} = \tan (θ)$

Step 3.1.2.3

Rewrite the expression.

$\frac{2}{t} = \tan (θ)$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t, 1$

Step 4.2

The LCM of one and any expression is the expression.

$t$

Step 5

Multiply each term in $\frac{2}{t} = \tan (θ)$ by $t$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{2}{t} = \tan (θ)$ by $t$ .

$\frac{2}{t} t = \tan (θ) t$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $t$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2}{t} t = \tan (θ) t$

Step 5.2.1.2

Rewrite the expression.

$2 = \tan (θ) t$

Step 5.3

Simplify the right side.

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Step 5.3.1

Reorder factors in $\tan (θ) t$ .

$2 = t \tan (θ)$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $t \tan (θ) = 2$ .

$t \tan (θ) = 2$

Step 6.2

Divide each term in $t \tan (θ) = 2$ by $\tan (θ)$ and simplify.

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Step 6.2.1

Divide each term in $t \tan (θ) = 2$ by $\tan (θ)$ .

$\frac{t \tan (θ)}{\tan (θ)} = \frac{2}{\tan (θ)}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $\tan (θ)$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{t \tan (θ)}{\tan (θ)} = \frac{2}{\tan (θ)}$

Step 6.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{2}{\tan (θ)}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Separate fractions.

$t = \frac{2}{1} \cdot \frac{1}{\tan (θ)}$

Step 6.2.3.2

Rewrite $\tan (θ)$ in terms of sines and cosines.

$t = \frac{2}{1} \cdot \frac{1}{\frac{\sin (θ)}{\cos (θ)}}$

Step 6.2.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

$t = \frac{2}{1} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 6.2.3.4

Convert from $\frac{\cos (θ)}{\sin (θ)}$ to $\cot (θ)$ .

$t = \frac{2}{1} \cot (θ)$

Step 6.2.3.5

Divide $2$ by $1$ .

$t = 2 \cot (θ)$