Trigonometry Examples

$\tan (θ) = \sin (θ)$

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

$1 = \frac{\sin (θ)}{\tan (θ)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

Cancel the common factor.

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

Step 1.3.4.2

Rewrite the expression.

$1 = \cos (θ)$

Step 2

Rewrite the equation as $\cos (θ) = 1$ .

$\cos (θ) = 1$

Step 3

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

$θ = \arccos (1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (1)$ is $0$ .

$θ = 0$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$θ = 2 π - 0$

Step 6

Subtract $0$ from $2 π$ .

$θ = 2 π$

Step 7

Find the period of $\cos (θ)$ .

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Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

The period of the $\cos (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$θ = 2 π n$ , for any integer $n$