Trigonometry Examples

7 tan (x) sin (x) = - 6 sin (x)

$7 \tan (x) \sin (x) = - 6 \sin (x)$

Step 1

Add

6 sin (x)

$6 \sin (x)$ to both sides of the equation.

7 tan (x) sin (x) + 6 sin (x) = 0

$7 \tan (x) \sin (x) + 6 \sin (x) = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite

tan (x)

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

7 (\frac{sin (x)}{cos (x)}) \cdot sin (x) + 6 sin (x) = 0

$7 (\frac{\sin (x)}{\cos (x)}) \cdot \sin (x) + 6 \sin (x) = 0$

Step 2.1.2

Combine

7

$7$ and

\frac{sin (x)}{cos (x)}

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

\frac{7 sin (x)}{cos (x)} \cdot sin (x) + 6 sin (x) = 0

$\frac{7 \sin (x)}{\cos (x)} \cdot \sin (x) + 6 \sin (x) = 0$

Step 2.1.3

Multiply

\frac{7 sin (x)}{cos (x)} sin (x)

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

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Step 2.1.3.1

Combine

\frac{7 sin (x)}{cos (x)}

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

sin (x)

$\sin (x)$ .

\frac{7 sin (x) sin (x)}{cos (x)} + 6 sin (x) = 0

$\frac{7 \sin (x) \sin (x)}{\cos (x)} + 6 \sin (x) = 0$

Step 2.1.3.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{7 (sin (x) sin (x))}{cos (x)} + 6 sin (x) = 0

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} + 6 \sin (x) = 0$

Step 2.1.3.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{7 (sin (x) sin (x))}{cos (x)} + 6 sin (x) = 0

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} + 6 \sin (x) = 0$

Step 2.1.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{7 {sin (x)}^{1 + 1}}{cos (x)} + 6 sin (x) = 0

$\frac{7 {\sin (x)}^{1 + 1}}{\cos (x)} + 6 \sin (x) = 0$

Step 2.1.3.5

Add

1

$1$ and

1

$1$ .

\frac{7 {sin}^{2} (x)}{cos (x)} + 6 sin (x) = 0

$\frac{7 \sin^{2} (x)}{\cos (x)} + 6 \sin (x) = 0$

\frac{7 {sin}^{2} (x)}{cos (x)} + 6 sin (x) = 0

$\frac{7 \sin^{2} (x)}{\cos (x)} + 6 \sin (x) = 0$

\frac{7 {sin}^{2} (x)}{cos (x)} + 6 sin (x) = 0

$\frac{7 \sin^{2} (x)}{\cos (x)} + 6 \sin (x) = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Factor

sin (x)

$\sin (x)$ out of

{sin}^{2} (x)

$\sin^{2} (x)$ .

\frac{7 (sin (x) sin (x))}{cos (x)} + 6 sin (x) = 0

$\frac{7 (\sin (x) \sin (x))}{\cos (x)} + 6 \sin (x) = 0$

Step 2.2.2

Separate fractions.

\frac{7 (sin (x))}{1} \cdot \frac{sin (x)}{cos (x)} + 6 sin (x) = 0

$\frac{7 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + 6 \sin (x) = 0$

Step 2.2.3

Convert from

\frac{sin (x)}{cos (x)}

$\frac{\sin (x)}{\cos (x)}$ to

tan (x)

$\tan (x)$ .

\frac{7 (sin (x))}{1} \cdot tan (x) + 6 sin (x) = 0

$\frac{7 (\sin (x))}{1} \cdot \tan (x) + 6 \sin (x) = 0$

Step 2.2.4

Divide

7 (sin (x))

$7 (\sin (x))$ by

1

$1$ .

7 sin (x) tan (x) + 6 sin (x) = 0

$7 \sin (x) \tan (x) + 6 \sin (x) = 0$

7 sin (x) tan (x) + 6 sin (x) = 0

$7 \sin (x) \tan (x) + 6 \sin (x) = 0$

7 sin (x) tan (x) + 6 sin (x) = 0

$7 \sin (x) \tan (x) + 6 \sin (x) = 0$

Step 3

Factor

sin (x)

$\sin (x)$ out of

7 sin (x) tan (x) + 6 sin (x)

$7 \sin (x) \tan (x) + 6 \sin (x)$ .

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Step 3.1

Factor

sin (x)

$\sin (x)$ out of

7 sin (x) tan (x)

$7 \sin (x) \tan (x)$ .

sin (x) (7 tan (x)) + 6 sin (x) = 0

$\sin (x) (7 \tan (x)) + 6 \sin (x) = 0$

Step 3.2

Factor

sin (x)

$\sin (x)$ out of

6 sin (x)

$6 \sin (x)$ .

sin (x) (7 tan (x)) + sin (x) \cdot 6 = 0

$\sin (x) (7 \tan (x)) + \sin (x) \cdot 6 = 0$

Step 3.3

Factor

sin (x)

$\sin (x)$ out of

sin (x) (7 tan (x)) + sin (x) \cdot 6

$\sin (x) (7 \tan (x)) + \sin (x) \cdot 6$ .

sin (x) (7 tan (x) + 6) = 0

$\sin (x) (7 \tan (x) + 6) = 0$

sin (x) (7 tan (x) + 6) = 0

$\sin (x) (7 \tan (x) + 6) = 0$

Step 4

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

sin (x) = 0

$\sin (x) = 0$

7 tan (x) + 6 = 0

$7 \tan (x) + 6 = 0$

Step 5

Set

sin (x)

$\sin (x)$ equal to

0

$0$ and solve for

x

$x$ .

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Step 5.1

Set

sin (x)

$\sin (x)$ equal to

0

$0$ .

sin (x) = 0

$\sin (x) = 0$

Step 5.2

Solve

sin (x) = 0

$\sin (x) = 0$ for

x

$x$ .

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Step 5.2.1

Take the inverse sine of both sides of the equation to extract

x

$x$ from inside the sine.

x = arcsin (0)

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of

arcsin (0)

$\arcsin (0)$ is

0

$0$ .

x = 0

$x = 0$

x = 0

$x = 0$

Step 5.2.3

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

180

$180$ to find the solution in the second quadrant.

x = 180 - 0

$x = 180 - 0$

Step 5.2.4

Subtract

0

$0$ from

180

$180$ .

x = 180

$x = 180$

Step 5.2.5

Find the period of

sin (x)

$\sin (x)$ .

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Step 5.2.5.1

The period of the function can be calculated using

\frac{360}{| b |}

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

\frac{360}{| b |}

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

Step 5.2.5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{360}{| 1 |}

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

Step 5.2.5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{360}{1}

$\frac{360}{1}$

Step 5.2.5.4

Divide

360

$360$ by

1

$1$ .

360

$360$

360

$360$

Step 5.2.6

The period of the

sin (x)

$\sin (x)$ function is

360

$360$ so values will repeat every

360

$360$ degrees in both directions.

x = 360 n, 180 + 360 n

$x = 360 n, 180 + 360 n$ , for any integer

n

$n$

x = 360 n, 180 + 360 n

$x = 360 n, 180 + 360 n$ , for any integer

n

$n$

x = 360 n, 180 + 360 n

$x = 360 n, 180 + 360 n$ , for any integer

n

$n$

Step 6

Set

7 tan (x) + 6

$7 \tan (x) + 6$ equal to

0

$0$ and solve for

x

$x$ .

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Step 6.1

Set

7 tan (x) + 6

$7 \tan (x) + 6$ equal to

0

$0$ .

7 tan (x) + 6 = 0

$7 \tan (x) + 6 = 0$

Step 6.2

Solve

7 tan (x) + 6 = 0

$7 \tan (x) + 6 = 0$ for

x

$x$ .

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Step 6.2.1

Subtract

6

$6$ from both sides of the equation.

7 tan (x) = - 6

$7 \tan (x) = - 6$

Step 6.2.2

Divide each term in

7 tan (x) = - 6

$7 \tan (x) = - 6$ by

7

$7$ and simplify.

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Step 6.2.2.1

Divide each term in

7 tan (x) = - 6

$7 \tan (x) = - 6$ by

7

$7$ .

\frac{7 tan (x)}{7} = \frac{- 6}{7}

$\frac{7 \tan (x)}{7} = \frac{- 6}{7}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of

7

$7$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{7 \tan (x)}{7} = \frac{- 6}{7}$

Step 6.2.2.2.1.2

Divide

$\tan (x)$ by

$1$ .

$\tan (x) = \frac{- 6}{7}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$\tan (x) = - \frac{6}{7}$

Step 6.2.3

Take the inverse tangent of both sides of the equation to extract

$x$ from inside the tangent.

$x = \arctan (- \frac{6}{7})$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Evaluate

$\arctan (- \frac{6}{7})$ .

$x = - 40.60129464$

Step 6.2.5

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from

$180$ to find the solution in the third quadrant.

$x = - 40.60129464 - 180$

Step 6.2.6

Simplify the expression to find the second solution.

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Step 6.2.6.1

Add

$360 °$ to

$- 40.60129464 - 180 °$ .

$x = - 40.60129464 - 180 ° + 360 °$

Step 6.2.6.2

The resulting angle of

$139.39870535 °$ is positive and coterminal with

$- 40.60129464 - 180$ .

$x = 139.39870535 °$

Step 6.2.7

Find the period of

$\tan (x)$ .

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Step 6.2.7.1

The period of the function can be calculated using

$\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 6.2.7.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 6.2.7.3

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

$0$ and

$1$ is

$1$ .

$\frac{180}{1}$

Step 6.2.7.4

Divide

$180$ by

$1$ .

$180$

Step 6.2.8

Add

$180$ to every negative angle to get positive angles.

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Step 6.2.8.1

Add

$180$ to

$- 40.60129464$ to find the positive angle.

$- 40.60129464 + 180$

Step 6.2.8.2

Subtract

$40.60129464$ from

$180$ .

$139.39870535$

Step 6.2.8.3

List the new angles.

$x = 139.39870535$

Step 6.2.9

The period of the

$\tan (x)$ function is

$180$ so values will repeat every

$180$ degrees in both directions.

$x = 139.39870535 + 180 n, 139.39870535 + 180 n$ , for any integer

$n$

$x = 139.39870535 + 180 n, 139.39870535 + 180 n$ , for any integer

$n$

$x = 139.39870535 + 180 n, 139.39870535 + 180 n$ , for any integer

$n$

Step 7

The final solution is all the values that make

$\sin (x) (7 \tan (x) + 6) = 0$ true.

$x = 360 n, 180 + 360 n, 139.39870535 + 180 n, 139.39870535 + 180 n$ , for any integer

$n$

Step 8

Consolidate the answers.

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Step 8.1

Consolidate

$360 n$ and

$180 + 360 n$ to

$180 n$ .

$x = 180 n, 139.39870535 + 180 n, 139.39870535 + 180 n$ , for any integer

$n$

Step 8.2

Consolidate

$139.39870535 + 180 n$ and

$139.39870535 + 180 n$ to

$139.39870535 + 180 n$ .

$x = 180 n, 139.39870535 + 180 n$ , for any integer

$n$

$x = 180 n, 139.39870535 + 180 n$ , for any integer

$n$