Trigonometry Examples

$2 \sin (θ) = \tan (θ)$

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Separate fractions.

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Write $\sin (θ)$ as a fraction with denominator $1$ .

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

Step 1.2.5

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

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

Rewrite the expression.

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

Step 1.2.6

Divide $2$ by $1$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

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

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Step 1.3.1.1

Cancel the common factor.

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

Step 1.3.1.2

Rewrite the expression.

$2 \cos (θ) = 1$

Step 2

Divide each term in $2 \cos (θ) = 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \cos (θ) = 1$ by $2$ .

$\frac{2 \cos (θ)}{2} = \frac{1}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \cos (θ)}{2} = \frac{1}{2}$

Step 2.2.1.2

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{1}{2}$

Step 3

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$θ = \arccos (\frac{1}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (\frac{1}{2})$ is $60$ .

$θ = 60$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$θ = 360 - 60$

Step 6

Subtract $60$ from $360$ .

$θ = 300$

Step 7

Find the period of $\cos (θ)$ .

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Step 7.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 7.4

Divide $360$ by $1$ .

$360$

Step 8

The period of the $\cos (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 60 + 360 n, 300 + 360 n$ , for any integer $n$