Trigonometry Examples

$5 \tan (x) \cos (x) = 7 \cos (x)$

Step 1

Simplify the left side.

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Step 1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1

Add parentheses.

$5 (\tan (x) \cos (x)) = 7 \cos (x)$

Step 1.1.2

Rewrite $5 \tan (x) \cos (x)$ in terms of sines and cosines.

$5 (\frac{\sin (x)}{\cos (x)} \cos (x)) = 7 \cos (x)$

Step 1.1.3

Cancel the common factors.

$5 \sin (x) = 7 \cos (x)$

Step 2

Divide each term in the equation by $\cos (x)$ .

$\frac{5 \sin (x)}{\cos (x)} = \frac{7 \cos (x)}{\cos (x)}$

Step 3

Separate fractions.

$\frac{5}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{7 \cos (x)}{\cos (x)}$

Step 4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{5}{1} \cdot \tan (x) = \frac{7 \cos (x)}{\cos (x)}$

Step 5

Divide $5$ by $1$ .

$5 \tan (x) = \frac{7 \cos (x)}{\cos (x)}$

Step 6

Cancel the common factor of $\cos (x)$ .

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Step 6.1

Cancel the common factor.

$5 \tan (x) = \frac{7 \cos (x)}{\cos (x)}$

Step 6.2

Divide $7$ by $1$ .

$5 \tan (x) = 7$

Step 7

Divide each term in $5 \tan (x) = 7$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 \tan (x) = 7$ by $5$ .

$\frac{5 \tan (x)}{5} = \frac{7}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 \tan (x)}{5} = \frac{7}{5}$

Step 7.2.1.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{7}{5}$

Step 8

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (\frac{7}{5})$

Step 9

Simplify the right side.

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Step 9.1

Evaluate $\arctan (\frac{7}{5})$ .

$x = 0.95054684$

Step 10

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 0.95054684$

Step 11

Solve for $x$ .

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Step 11.1

Remove parentheses.

$x = 3.14159265 + 0.95054684$

Step 11.2

Remove parentheses.

$x = (3.14159265) + 0.95054684$

Step 11.3

Add $3.14159265$ and $0.95054684$ .

$x = 4.09213949$

Step 12

Find the period of $\tan (x)$ .

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Step 12.1

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

$\frac{π}{| b |}$

Step 12.2

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

$\frac{π}{| 1 |}$

Step 12.3

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

$\frac{π}{1}$

Step 12.4

Divide $π$ by $1$ .

$π$

Step 13

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

$x = 0.95054684 + π n, 4.09213949 + π n$ , for any integer $n$

Step 14

Consolidate $0.95054684 + π n$ and $4.09213949 + π n$ to $0.95054684 + π n$ .

$x = 0.95054684 + π n$ , for any integer $n$