Trigonometry Examples

$\cos (\frac{π}{2} + x) - \sin (\frac{π}{2} + x) = 0$

Step 1

Divide each term in the equation by $\cos (\frac{π}{2} + x)$ .

$\frac{\cos (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} + \frac{- \sin (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 2

Cancel the common factor of $\cos (\frac{π}{2} + x)$ .

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Step 2.1

Cancel the common factor.

$\frac{\cos (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} + \frac{- \sin (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 2.2

Rewrite the expression.

$1 + \frac{- \sin (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 3

Separate fractions.

$1 + \frac{- 1}{1} \cdot \frac{\sin (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)} = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 4

Convert from $\frac{\sin (\frac{π}{2} + x)}{\cos (\frac{π}{2} + x)}$ to $\tan (\frac{π}{2} + x)$ .

$1 + \frac{- 1}{1} \cdot \tan (\frac{π}{2} + x) = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 5

Divide $- 1$ by $1$ .

$1 - \tan (\frac{π}{2} + x) = \frac{0}{\cos (\frac{π}{2} + x)}$

Step 6

Separate fractions.

$1 - \tan (\frac{π}{2} + x) = \frac{0}{1} \cdot \frac{1}{\cos (\frac{π}{2} + x)}$

Step 7

Convert from $\frac{1}{\cos (\frac{π}{2} + x)}$ to $\sec (\frac{π}{2} + x)$ .

$1 - \tan (\frac{π}{2} + x) = \frac{0}{1} \cdot \sec (\frac{π}{2} + x)$

Step 8

Divide $0$ by $1$ .

$1 - \tan (\frac{π}{2} + x) = 0 \sec (\frac{π}{2} + x)$

Step 9

Multiply $0$ by $\sec (\frac{π}{2} + x)$ .

$1 - \tan (\frac{π}{2} + x) = 0$

Step 10

Subtract $1$ from both sides of the equation.

$- \tan (\frac{π}{2} + x) = - 1$

Step 11

Divide each term in $- \tan (\frac{π}{2} + x) = - 1$ by $- 1$ and simplify.

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Step 11.1

Divide each term in $- \tan (\frac{π}{2} + x) = - 1$ by $- 1$ .

$\frac{- \tan (\frac{π}{2} + x)}{- 1} = \frac{- 1}{- 1}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (\frac{π}{2} + x)}{1} = \frac{- 1}{- 1}$

Step 11.2.2

Divide $\tan (\frac{π}{2} + x)$ by $1$ .

$\tan (\frac{π}{2} + x) = \frac{- 1}{- 1}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Divide $- 1$ by $- 1$ .

$\tan (\frac{π}{2} + x) = 1$

Step 12

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

$\frac{π}{2} + x = \arctan (1)$

Step 13

Simplify the right side.

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Step 13.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$\frac{π}{2} + x = \frac{π}{4}$

Step 14

Move all terms not containing $x$ to the right side of the equation.

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Step 14.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = \frac{π}{4} - \frac{π}{2}$

Step 14.2

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

$x = \frac{π}{4} - \frac{π}{2} \cdot \frac{2}{2}$

Step 14.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$x = \frac{π}{4} - \frac{π \cdot 2}{2 \cdot 2}$

Step 14.3.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} - \frac{π \cdot 2}{4}$

Step 14.4

Combine the numerators over the common denominator.

$x = \frac{π - π \cdot 2}{4}$

Step 14.5

Simplify the numerator.

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Step 14.5.1

Multiply $2$ by $- 1$ .

$x = \frac{π - 2 π}{4}$

Step 14.5.2

Subtract $2 π$ from $π$ .

$x = \frac{- π}{4}$

Step 14.6

Move the negative in front of the fraction.

$x = - \frac{π}{4}$

Step 15

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.

$\frac{π}{2} + x = π + \frac{π}{4}$

Step 16

Solve for $x$ .

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Step 16.1

Simplify $π + \frac{π}{4}$ .

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Step 16.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π}{2} + x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 16.1.2

Combine fractions.

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Step 16.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π}{2} + x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 16.1.2.2

Combine the numerators over the common denominator.

$\frac{π}{2} + x = \frac{π \cdot 4 + π}{4}$

Step 16.1.3

Simplify the numerator.

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Step 16.1.3.1

Move $4$ to the left of $π$ .

$\frac{π}{2} + x = \frac{4 \cdot π + π}{4}$

Step 16.1.3.2

Add $4 π$ and $π$ .

$\frac{π}{2} + x = \frac{5 π}{4}$

Step 16.2

Move all terms not containing $x$ to the right side of the equation.

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Step 16.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = \frac{5 π}{4} - \frac{π}{2}$

Step 16.2.2

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

$x = \frac{5 π}{4} - \frac{π}{2} \cdot \frac{2}{2}$

Step 16.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.2.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$x = \frac{5 π}{4} - \frac{π \cdot 2}{2 \cdot 2}$

Step 16.2.3.2

Multiply $2$ by $2$ .

$x = \frac{5 π}{4} - \frac{π \cdot 2}{4}$

Step 16.2.4

Combine the numerators over the common denominator.

$x = \frac{5 π - π \cdot 2}{4}$

Step 16.2.5

Simplify the numerator.

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Step 16.2.5.1

Multiply $2$ by $- 1$ .

$x = \frac{5 π - 2 π}{4}$

Step 16.2.5.2

Subtract $2 π$ from $5 π$ .

$x = \frac{3 π}{4}$

Step 17

Find the period of $\tan (\frac{π}{2} + x)$ .

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Step 17.1

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

$\frac{π}{| b |}$

Step 17.2

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

$\frac{π}{| 1 |}$

Step 17.3

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

$\frac{π}{1}$

Step 17.4

Divide $π$ by $1$ .

$π$

Step 18

Add $π$ to every negative angle to get positive angles.

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Step 18.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 18.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 18.3

Combine fractions.

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Step 18.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 18.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 18.4

Simplify the numerator.

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Step 18.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 18.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 18.5

List the new angles.

$x = \frac{3 π}{4}$

Step 19

The period of the $\tan (\frac{π}{2} + x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 20

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$