Trigonometry Examples

$4 \sin (x) \cos (x) + \cos (x) = 0$

Step 1

Factor

$\cos (x)$ out of

$4 \sin (x) \cos (x) + \cos (x)$ .

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Step 1.1

Factor

$\cos (x)$ out of

$4 \sin (x) \cos (x)$ .

$\cos (x) (4 \sin (x)) + \cos (x) = 0$

Step 1.2

Raise

$\cos (x)$ to the power of

$1$ .

$\cos (x) (4 \sin (x)) + \cos (x) = 0$

Step 1.3

Factor

$\cos (x)$ out of

$\cos^{1} (x)$ .

$\cos (x) (4 \sin (x)) + \cos (x) \cdot 1 = 0$

Step 1.4

Factor

$\cos (x)$ out of

$\cos (x) (4 \sin (x)) + \cos (x) \cdot 1$ .

$\cos (x) (4 \sin (x) + 1) = 0$

Step 2

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

$0$ , the entire expression will be equal to

$0$ .

$\cos (x) = 0$

$4 \sin (x) + 1 = 0$

Step 3

Set

$\cos (x)$ equal to

$0$ and solve for

$x$ .

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Step 3.1

Set

$\cos (x)$ equal to

$0$ .

$\cos (x) = 0$

Step 3.2

Solve

$\cos (x) = 0$ for

$x$ .

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Step 3.2.1

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

$x$ from inside the cosine.

$x = \arccos (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

The exact value of

$\arccos (0)$ is

$90$ .

$x = 90$

Step 3.2.3

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.

$x = 360 - 90$

Step 3.2.4

Subtract

$90$ from

$360$ .

$x = 270$

Step 3.2.5

Find the period of

$\cos (x)$ .

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Step 3.2.5.1

The period of the function can be calculated using

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

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

Step 3.2.5.2

Replace

$b$ with

$1$ in the formula for period.

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

Step 3.2.5.3

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

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 3.2.5.4

Divide

$360$ by

$1$ .

$360$

Step 3.2.6

The period of the

$\cos (x)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$x = 90 + 360 n, 270 + 360 n$ , for any integer

$n$

$x = 90 + 360 n, 270 + 360 n$ , for any integer

$n$

$x = 90 + 360 n, 270 + 360 n$ , for any integer

$n$

Step 4

Set

$4 \sin (x) + 1$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$4 \sin (x) + 1$ equal to

$0$ .

$4 \sin (x) + 1 = 0$

Step 4.2

Solve

$4 \sin (x) + 1 = 0$ for

$x$ .

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Step 4.2.1

Subtract

$1$ from both sides of the equation.

$4 \sin (x) = - 1$

Step 4.2.2

Divide each term in

$4 \sin (x) = - 1$ by

$4$ and simplify.

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Step 4.2.2.1

Divide each term in

$4 \sin (x) = - 1$ by

$4$ .

$\frac{4 \sin (x)}{4} = \frac{- 1}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

$4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 \sin (x)}{4} = \frac{- 1}{4}$

Step 4.2.2.2.1.2

Divide

$\sin (x)$ by

$1$ .

$\sin (x) = \frac{- 1}{4}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{4}$

Step 4.2.3

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

$x$ from inside the sine.

$x = \arcsin (- \frac{1}{4})$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Evaluate

$\arcsin (- \frac{1}{4})$ .

$x = - 14.47751218$

Step 4.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$x = 360 + 14.47751218 + 180$

Step 4.2.6

Simplify the expression to find the second solution.

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Step 4.2.6.1

Subtract

$360 °$ from

$360 + 14.47751218 + 180 °$ .

$x = 360 + 14.47751218 + 180 ° - 360 °$

Step 4.2.6.2

The resulting angle of

$194.47751218 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 14.47751218 + 180$ .

$x = 194.47751218 °$

Step 4.2.7

Find the period of

$\sin (x)$ .

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Step 4.2.7.1

The period of the function can be calculated using

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

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

Step 4.2.7.2

Replace

$b$ with

$1$ in the formula for period.

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

Step 4.2.7.3

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

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 4.2.7.4

Divide

$360$ by

$1$ .

$360$

Step 4.2.8

Add

$360$ to every negative angle to get positive angles.

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Step 4.2.8.1

Add

$360$ to

$- 14.47751218$ to find the positive angle.

$- 14.47751218 + 360$

Step 4.2.8.2

Subtract

$14.47751218$ from

$360$ .

$345.52248781$

Step 4.2.8.3

List the new angles.

$x = 345.52248781$

Step 4.2.9

The period of the

$\sin (x)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$x = 194.47751218 + 360 n, 345.52248781 + 360 n$ , for any integer

$n$

$x = 194.47751218 + 360 n, 345.52248781 + 360 n$ , for any integer

$n$

$x = 194.47751218 + 360 n, 345.52248781 + 360 n$ , for any integer

$n$

Step 5

The final solution is all the values that make

$\cos (x) (4 \sin (x) + 1) = 0$ true.

$x = 90 + 360 n, 270 + 360 n, 194.47751218 + 360 n, 345.52248781 + 360 n$ , for any integer

$n$

Step 6

Consolidate

$90 + 360 n$ and

$270 + 360 n$ to

$90 + 180 n$ .

$x = 90 + 180 n, 194.47751218 + 360 n, 345.52248781 + 360 n$ , for any integer

$n$