Trigonometry Examples

$\sin (2 x) - 2 \cos (x) = 0$ , $(0, 2 π)$

Step 1

Apply the sine double-angle identity.

$2 \sin (x) \cos (x) - 2 \cos (x) = 0$

Step 2

Factor $2 \cos (x)$ out of $2 \sin (x) \cos (x) - 2 \cos (x)$ .

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Step 2.1

Factor $2 \cos (x)$ out of $2 \sin (x) \cos (x)$ .

$2 \cos (x) \sin (x) - 2 \cos (x) = 0$

Step 2.2

Factor $2 \cos (x)$ out of $- 2 \cos (x)$ .

$2 \cos (x) \sin (x) + 2 \cos (x) \cdot - 1 = 0$

Step 2.3

Factor $2 \cos (x)$ out of $2 \cos (x) \sin (x) + 2 \cos (x) \cdot - 1$ .

$2 \cos (x) (\sin (x) - 1) = 0$

Step 3

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

$\cos (x) = 0$

$\sin (x) - 1 = 0$

Step 4

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

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Step 4.1

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

$\cos (x) = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \arccos (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 4.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 4.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 4.2.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 4.2.4.2

Combine fractions.

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Step 4.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 4.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 4.2.4.3

Simplify the numerator.

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Step 4.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 4.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 4.2.5

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

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Step 4.2.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.5.3

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

$\frac{2 π}{1}$

Step 4.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.6

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 5

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Step 5.2

Solve $\sin (x) - 1 = 0$ for $x$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$\sin (x) = 1$

Step 5.2.2

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

$x = \arcsin (1)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 5.2.4

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.

$x = π - \frac{π}{2}$

Step 5.2.5

Simplify $π - \frac{π}{2}$ .

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Step 5.2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 5.2.5.2

Combine fractions.

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Step 5.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 5.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 5.2.5.3

Simplify the numerator.

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Step 5.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 5.2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 5.2.6

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

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Step 5.2.6.1

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

$\frac{2 π}{| b |}$

Step 5.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.6.3

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

$\frac{2 π}{1}$

Step 5.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7

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

$x = \frac{π}{2} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $2 \cos (x) (\sin (x) - 1) = 0$ true.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 7

Consolidate the answers.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 8

Find the values of $n$ that produce a value within the interval $(0, 2 π)$ .

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Step 8.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 8.1.1

Plug in $0$ for $n$ .

$\frac{π}{2} + π (0)$

Step 8.1.2

Simplify.

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Step 8.1.2.1

Multiply $π$ by $0$ .

$\frac{π}{2} + 0$

Step 8.1.2.2

Add $\frac{π}{2}$ and $0$ .

$\frac{π}{2}$

Step 8.1.3

The interval $(0, 2 π)$ contains $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 8.2

Plug in $1$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 8.2.1

Plug in $1$ for $n$ .

$\frac{π}{2} + π (1)$

Step 8.2.2

Simplify.

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Step 8.2.2.1

Multiply $π$ by $1$ .

$\frac{π}{2} + π$

Step 8.2.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π}{2} + π \cdot \frac{2}{2}$

Step 8.2.2.3

Combine fractions.

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Step 8.2.2.3.1

Combine $π$ and $\frac{2}{2}$ .

$\frac{π}{2} + \frac{π \cdot 2}{2}$

Step 8.2.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 2}{2}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Move $2$ to the left of $π$ .

$\frac{π + 2 \cdot π}{2}$

Step 8.2.2.4.2

Add $π$ and $2 π$ .

$\frac{3 π}{2}$

Step 8.2.3

The interval $(0, 2 π)$ contains $\frac{3 π}{2}$ .

$x = \frac{π}{2}, \frac{3 π}{2}$