Trigonometry Examples

$\cos (x) \csc (x) = 2 \cos (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\cos (x) \csc (x)$ .

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Step 1.1.1

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

$\cos (x) \frac{1}{\sin (x)} = 2 \cos (x)$

Step 1.1.2

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

$\frac{\cos (x)}{\sin (x)} = 2 \cos (x)$

Step 2

Multiply both sides of the equation by $\sin (x)$ .

$\sin (x) \frac{\cos (x)}{\sin (x)} = \sin (x) (2 \cos (x))$

Step 3

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

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Step 3.1

Cancel the common factor.

$\sin (x) \frac{\cos (x)}{\sin (x)} = \sin (x) (2 \cos (x))$

Step 3.2

Rewrite the expression.

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

Apply the sine double-angle identity.

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

Step 6

Subtract $\sin (2 x)$ from both sides of the equation.

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

Step 7

Simplify each term.

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Step 7.1

Apply the sine double-angle identity.

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

Step 7.2

Multiply $2$ by $- 1$ .

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

Step 8

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

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Step 8.1

Factor $\cos (x)$ out of $\cos^{1} (x)$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

$\cos (x) = 0$

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

Step 10

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

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Step 10.1

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

$\cos (x) = 0$

Step 10.2

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

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Step 10.2.1

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

$x = \arccos (0)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

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

$x = \frac{π}{2}$

Step 10.2.3

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 10.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 10.2.4.1

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 10.2.4.2

Combine fractions.

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Step 10.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 10.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 10.2.4.3

Simplify the numerator.

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Step 10.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 10.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 10.2.5

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

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Step 10.2.5.1

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

$\frac{2 π}{| b |}$

Step 10.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 10.2.5.3

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

$\frac{2 π}{1}$

Step 10.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.2.6

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 11

Set $1 - 2 \sin (x)$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $1 - 2 \sin (x)$ equal to $0$ .

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

Step 11.2

Solve $1 - 2 \sin (x) = 0$ for $x$ .

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Step 11.2.1

Subtract $1$ from both sides of the equation.

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

Step 11.2.2

Divide each term in $- 2 \sin (x) = - 1$ by $- 2$ and simplify.

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Step 11.2.2.1

Divide each term in $- 2 \sin (x) = - 1$ by $- 2$ .

$\frac{- 2 \sin (x)}{- 2} = \frac{- 1}{- 2}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (x)}{- 2} = \frac{- 1}{- 2}$

Step 11.2.2.2.1.2

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

$\sin (x) = \frac{- 1}{- 2}$

Step 11.2.2.3

Simplify the right side.

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Step 11.2.2.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{1}{2}$

Step 11.2.3

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

$x = \arcsin (\frac{1}{2})$

Step 11.2.4

Simplify the right side.

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Step 11.2.4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 11.2.5

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.

$x = π - \frac{π}{6}$

Step 11.2.6

Simplify $π - \frac{π}{6}$ .

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Step 11.2.6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 11.2.6.2

Combine fractions.

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Step 11.2.6.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 11.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 11.2.6.3

Simplify the numerator.

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Step 11.2.6.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 11.2.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 11.2.7

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

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Step 11.2.7.1

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

$\frac{2 π}{| b |}$

Step 11.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 11.2.7.3

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

$\frac{2 π}{1}$

Step 11.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.2.8

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 12

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 13

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{2} + π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$