Trigonometry Examples

$7 \tan (x) \sin (x) - 4 \tan (x) = 0$

Step 1

Simplify the left side of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$7 (\frac{\sin (x)}{\cos (x)}) \cdot \sin (x) - 4 \tan (x) = 0$

Step 1.1.2

Combine $7$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{7 \sin (x)}{\cos (x)} \cdot \sin (x) - 4 \tan (x) = 0$

Step 1.1.3

Multiply $\frac{7 \sin (x)}{\cos (x)} \sin (x)$ .

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Step 1.1.3.1

Combine $\frac{7 \sin (x)}{\cos (x)}$ and $\sin (x)$ .

$\frac{7 \sin (x) \sin (x)}{\cos (x)} - 4 \tan (x) = 0$

Step 1.1.3.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} - 4 \tan (x) = 0$

Step 1.1.3.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} - 4 \tan (x) = 0$

Step 1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{7 {\sin (x)}^{1 + 1}}{\cos (x)} - 4 \tan (x) = 0$

Step 1.1.3.5

Add $1$ and $1$ .

$\frac{7 \sin^{2} (x)}{\cos (x)} - 4 \tan (x) = 0$

Step 1.1.4

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

$\frac{7 \sin^{2} (x)}{\cos (x)} - 4 \frac{\sin (x)}{\cos (x)} = 0$

Step 1.1.5

Combine $- 4$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{7 \sin^{2} (x)}{\cos (x)} + \frac{- 4 \sin (x)}{\cos (x)} = 0$

Step 1.1.6

Move the negative in front of the fraction.

$\frac{7 \sin^{2} (x)}{\cos (x)} - \frac{4 \sin (x)}{\cos (x)} = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} - \frac{4 \sin (x)}{\cos (x)} = 0$

Step 1.2.2

Separate fractions.

$\frac{7 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} - \frac{4 \sin (x)}{\cos (x)} = 0$

Step 1.2.3

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

$\frac{7 (\sin (x))}{1} \cdot \tan (x) - \frac{4 \sin (x)}{\cos (x)} = 0$

Step 1.2.4

Divide $7 (\sin (x))$ by $1$ .

$7 (\sin (x)) \tan (x) - \frac{4 \sin (x)}{\cos (x)} = 0$

Step 1.2.5

Separate fractions.

$7 \sin (x) \tan (x) - (\frac{4}{1} \cdot \frac{\sin (x)}{\cos (x)}) = 0$

Step 1.2.6

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

$7 \sin (x) \tan (x) - (\frac{4}{1} \cdot \tan (x)) = 0$

Step 1.2.7

Divide $4$ by $1$ .

$7 \sin (x) \tan (x) - (4 \tan (x)) = 0$

Step 1.2.8

Multiply $4$ by $- 1$ .

$7 \sin (x) \tan (x) - 4 \tan (x) = 0$

Step 2

Factor $\tan (x)$ out of $7 \sin (x) \tan (x) - 4 \tan (x)$ .

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Step 2.1

Factor $\tan (x)$ out of $7 \sin (x) \tan (x)$ .

$\tan (x) (7 \sin (x)) - 4 \tan (x) = 0$

Step 2.2

Factor $\tan (x)$ out of $- 4 \tan (x)$ .

$\tan (x) (7 \sin (x)) + \tan (x) \cdot - 4 = 0$

Step 2.3

Factor $\tan (x)$ out of $\tan (x) (7 \sin (x)) + \tan (x) \cdot - 4$ .

$\tan (x) (7 \sin (x) - 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (x) = 0$

$7 \sin (x) - 4 = 0$

Step 4

Set $\tan (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\tan (x)$ equal to $0$ .

$\tan (x) = 0$

Step 4.2

Solve $\tan (x) = 0$ for $x$ .

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Step 4.2.1

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

$x = \arctan (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

Step 4.2.3

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.

$x = π + 0$

Step 4.2.4

Add $π$ and $0$ .

$x = π$

Step 4.2.5

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

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Step 4.2.5.1

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

$\frac{π}{| b |}$

Step 4.2.5.2

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

$\frac{π}{| 1 |}$

Step 4.2.5.3

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

$\frac{π}{1}$

Step 4.2.5.4

Divide $π$ by $1$ .

$π$

Step 4.2.6

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

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

Step 5

Set $7 \sin (x) - 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $7 \sin (x) - 4$ equal to $0$ .

$7 \sin (x) - 4 = 0$

Step 5.2

Solve $7 \sin (x) - 4 = 0$ for $x$ .

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Step 5.2.1

Add $4$ to both sides of the equation.

$7 \sin (x) = 4$

Step 5.2.2

Divide each term in $7 \sin (x) = 4$ by $7$ and simplify.

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Step 5.2.2.1

Divide each term in $7 \sin (x) = 4$ by $7$ .

$\frac{7 \sin (x)}{7} = \frac{4}{7}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{7 \sin (x)}{7} = \frac{4}{7}$

Step 5.2.2.2.1.2

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

$\sin (x) = \frac{4}{7}$

Step 5.2.3

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

$x = \arcsin (\frac{4}{7})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

Evaluate $\arcsin (\frac{4}{7})$ .

$x = 0.60824557$

Step 5.2.5

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

$x = (3.14159265) - 0.60824557$

Step 5.2.6

Solve for $x$ .

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Step 5.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.60824557$

Step 5.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.60824557$

Step 5.2.6.3

Subtract $0.60824557$ from $3.14159265$ .

$x = 2.53334707$

Step 5.2.7

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

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Step 5.2.7.1

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

$\frac{2 π}{| b |}$

Step 5.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

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

$\frac{2 π}{1}$

Step 5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8

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

$x = 0.60824557 + 2 π n, 2.53334707 + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $\tan (x) (7 \sin (x) - 4) = 0$ true.

$x = π n, π + π n, 0.60824557 + 2 π n, 2.53334707 + 2 π n$ , for any integer $n$

Step 7

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

$x = π n, 0.60824557 + 2 π n, 2.53334707 + 2 π n$ , for any integer $n$