Precalculus Examples

$y = - \cos (\frac{1}{2} x + \frac{π}{5})$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \cos (\frac{1}{2} x + \frac{π}{5})$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- \cos (\frac{1}{2} x + \frac{π}{5}) = 0$ .

$- \cos (\frac{1}{2} x + \frac{π}{5}) = 0$

Step 1.2.2

Divide each term in $- \cos (\frac{1}{2} x + \frac{π}{5}) = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $- \cos (\frac{1}{2} x + \frac{π}{5}) = 0$ by $- 1$ .

$\frac{- \cos (\frac{1}{2} x + \frac{π}{5})}{- 1} = \frac{0}{- 1}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (\frac{1}{2} x + \frac{π}{5})}{1} = \frac{0}{- 1}$

Step 1.2.2.2.2

Divide $\cos (\frac{1}{2} x + \frac{π}{5})$ by $1$ .

$\cos (\frac{1}{2} x + \frac{π}{5}) = \frac{0}{- 1}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $0$ by $- 1$ .

$\cos (\frac{1}{2} x + \frac{π}{5}) = 0$

Step 1.2.3

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

$\frac{1}{2} x + \frac{π}{5} = \arccos (0)$

Step 1.2.4

Simplify the left side.

Tap for more steps...

Step 1.2.4.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{π}{5} = \arccos (0)$

Step 1.2.5

Simplify the right side.

Tap for more steps...

Step 1.2.5.1

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

$\frac{x}{2} + \frac{π}{5} = \frac{π}{2}$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

Subtract $\frac{π}{5}$ from both sides of the equation.

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

Step 1.2.6.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.2.6.3

To write $- \frac{π}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.6.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.6.4.1

Multiply $\frac{π}{2}$ by $\frac{5}{5}$ .

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

Step 1.2.6.4.2

Multiply $2$ by $5$ .

$\frac{x}{2} = \frac{π \cdot 5}{10} - \frac{π}{5} \cdot \frac{2}{2}$

Step 1.2.6.4.3

Multiply $\frac{π}{5}$ by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{π \cdot 5}{10} - \frac{π \cdot 2}{5 \cdot 2}$

Step 1.2.6.4.4

Multiply $5$ by $2$ .

$\frac{x}{2} = \frac{π \cdot 5}{10} - \frac{π \cdot 2}{10}$

Step 1.2.6.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{π \cdot 5 - π \cdot 2}{10}$

Step 1.2.6.6

Simplify the numerator.

Tap for more steps...

Step 1.2.6.6.1

Move $5$ to the left of $π$ .

$\frac{x}{2} = \frac{5 \cdot π - π \cdot 2}{10}$

Step 1.2.6.6.2

Multiply $2$ by $- 1$ .

$\frac{x}{2} = \frac{5 π - 2 π}{10}$

Step 1.2.6.6.3

Subtract $2 π$ from $5 π$ .

$\frac{x}{2} = \frac{3 π}{10}$

Step 1.2.7

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{3 π}{10}$

Step 1.2.8

Simplify both sides of the equation.

Tap for more steps...

Step 1.2.8.1

Simplify the left side.

Tap for more steps...

Step 1.2.8.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.8.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{3 π}{10}$

Step 1.2.8.1.1.2

Rewrite the expression.

$x = 2 \frac{3 π}{10}$

Step 1.2.8.2

Simplify the right side.

Tap for more steps...

Step 1.2.8.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.8.2.1.1

Factor $2$ out of $10$ .

$x = 2 \frac{3 π}{2 (5)}$

Step 1.2.8.2.1.2

Cancel the common factor.

$x = 2 \frac{3 π}{2 \cdot 5}$

Step 1.2.8.2.1.3

Rewrite the expression.

$x = \frac{3 π}{5}$

Step 1.2.9

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.

$\frac{x}{2} + \frac{π}{5} = 2 π - \frac{π}{2}$

Step 1.2.10

Solve for $x$ .

Tap for more steps...

Step 1.2.10.1

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 1.2.10.1.1

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

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

Step 1.2.10.1.2

Combine fractions.

Tap for more steps...

Step 1.2.10.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.10.1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.10.1.3

Simplify the numerator.

Tap for more steps...

Step 1.2.10.1.3.1

Multiply $2$ by $2$ .

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

Step 1.2.10.1.3.2

Subtract $π$ from $4 π$ .

$\frac{x}{2} + \frac{π}{5} = \frac{3 π}{2}$

Step 1.2.10.2

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

Tap for more steps...

Step 1.2.10.2.1

Subtract $\frac{π}{5}$ from both sides of the equation.

$\frac{x}{2} = \frac{3 π}{2} - \frac{π}{5}$

Step 1.2.10.2.2

To write $\frac{3 π}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{2} = \frac{3 π}{2} \cdot \frac{5}{5} - \frac{π}{5}$

Step 1.2.10.2.3

To write $- \frac{π}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3 π}{2} \cdot \frac{5}{5} - \frac{π}{5} \cdot \frac{2}{2}$

Step 1.2.10.2.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.10.2.4.1

Multiply $\frac{3 π}{2}$ by $\frac{5}{5}$ .

$\frac{x}{2} = \frac{3 π \cdot 5}{2 \cdot 5} - \frac{π}{5} \cdot \frac{2}{2}$

Step 1.2.10.2.4.2

Multiply $2$ by $5$ .

$\frac{x}{2} = \frac{3 π \cdot 5}{10} - \frac{π}{5} \cdot \frac{2}{2}$

Step 1.2.10.2.4.3

Multiply $\frac{π}{5}$ by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3 π \cdot 5}{10} - \frac{π \cdot 2}{5 \cdot 2}$

Step 1.2.10.2.4.4

Multiply $5$ by $2$ .

$\frac{x}{2} = \frac{3 π \cdot 5}{10} - \frac{π \cdot 2}{10}$

Step 1.2.10.2.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{3 π \cdot 5 - π \cdot 2}{10}$

Step 1.2.10.2.6

Simplify the numerator.

Tap for more steps...

Step 1.2.10.2.6.1

Multiply $5$ by $3$ .

$\frac{x}{2} = \frac{15 π - π \cdot 2}{10}$

Step 1.2.10.2.6.2

Multiply $2$ by $- 1$ .

$\frac{x}{2} = \frac{15 π - 2 π}{10}$

Step 1.2.10.2.6.3

Subtract $2 π$ from $15 π$ .

$\frac{x}{2} = \frac{13 π}{10}$

Step 1.2.10.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{13 π}{10}$

Step 1.2.10.4

Simplify both sides of the equation.

Tap for more steps...

Step 1.2.10.4.1

Simplify the left side.

Tap for more steps...

Step 1.2.10.4.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.10.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{13 π}{10}$

Step 1.2.10.4.1.1.2

Rewrite the expression.

$x = 2 \frac{13 π}{10}$

Step 1.2.10.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.10.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.10.4.2.1.1

Factor $2$ out of $10$ .

$x = 2 \frac{13 π}{2 (5)}$

Step 1.2.10.4.2.1.2

Cancel the common factor.

$x = 2 \frac{13 π}{2 \cdot 5}$

Step 1.2.10.4.2.1.3

Rewrite the expression.

$x = \frac{13 π}{5}$

Step 1.2.11

Find the period of $\cos (\frac{1}{2} x + \frac{π}{5})$ .

Tap for more steps...

Step 1.2.11.1

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

$\frac{2 π}{| b |}$

Step 1.2.11.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 1.2.11.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 1.2.11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 1.2.11.5

Multiply $2$ by $2$ .

$4 π$

Step 1.2.12

The period of the $\cos (\frac{1}{2} x + \frac{π}{5})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{3 π}{5} + 4 π n, \frac{13 π}{5} + 4 π n$ , for any integer $n$

Step 1.2.13

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3 π}{5} + 2 π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \cos (\frac{1}{2} \cdot (0) + \frac{π}{5})$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Multiply $\frac{1}{2}$ by $0$ .

$y = - \cos (\frac{1}{2} \cdot 0 + \frac{π}{5})$

Step 2.2.2

Remove parentheses.

$y = - \cos (\frac{1}{2} \cdot (0) + \frac{π}{5})$

Step 2.2.3

Simplify $- \cos (\frac{1}{2} \cdot (0) + \frac{π}{5})$ .

Tap for more steps...

Step 2.2.3.1

Multiply $\frac{1}{2}$ by $0$ .

$y = - \cos (0 + \frac{π}{5})$

Step 2.2.3.2

Add $0$ and $\frac{π}{5}$ .

$y = - \cos (\frac{π}{5})$

Step 2.2.3.3

Evaluate $\cos (\frac{π}{5})$ .

$y = - 1 \cdot 0.80901699$

Step 2.2.3.4

Multiply $- 1$ by $0.80901699$ .

$y = - 0.80901699$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 0.80901699)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3 π}{5} + 2 π n, 0)$ , for any integer $n$

y-intercept(s): $(0, - 0.80901699)$

Step 4