Trigonometry Examples

$f (x) = - 4 \cos (x - \frac{π}{2})$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = - 4 \cos (x - \frac{π}{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4 \cos (x - \frac{π}{2}) = 0$ .

$- 4 \cos (x - \frac{π}{2}) = 0$

Step 1.2.2

Divide each term in $- 4 \cos (x - \frac{π}{2}) = 0$ by $- 4$ and simplify.

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Step 1.2.2.1

Divide each term in $- 4 \cos (x - \frac{π}{2}) = 0$ by $- 4$ .

$\frac{- 4 \cos (x - \frac{π}{2})}{- 4} = \frac{0}{- 4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 \cos (x - \frac{π}{2})}{- 4} = \frac{0}{- 4}$

Step 1.2.2.2.1.2

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

$\cos (x - \frac{π}{2}) = \frac{0}{- 4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 4$ .

$\cos (x - \frac{π}{2}) = 0$

Step 1.2.3

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

$x - \frac{π}{2} = \arccos (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$x - \frac{π}{2} = \frac{π}{2}$

Step 1.2.5

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

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Step 1.2.5.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2} + \frac{π}{2}$

Step 1.2.5.2

Combine the numerators over the common denominator.

$x = \frac{π + π}{2}$

Step 1.2.5.3

Add $π$ and $π$ .

$x = \frac{2 π}{2}$

Step 1.2.5.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 1.2.5.4.2

Divide $π$ by $1$ .

$x = π$

Step 1.2.6

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.

$x - \frac{π}{2} = 2 π - \frac{π}{2}$

Step 1.2.7

Solve for $x$ .

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Step 1.2.7.1

Simplify $2 π - \frac{π}{2}$ .

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Step 1.2.7.1.1

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

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

Step 1.2.7.1.2

Combine fractions.

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Step 1.2.7.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.7.1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.7.1.3

Simplify the numerator.

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Step 1.2.7.1.3.1

Multiply $2$ by $2$ .

$x - \frac{π}{2} = \frac{4 π - π}{2}$

Step 1.2.7.1.3.2

Subtract $π$ from $4 π$ .

$x - \frac{π}{2} = \frac{3 π}{2}$

Step 1.2.7.2

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

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Step 1.2.7.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{3 π}{2} + \frac{π}{2}$

Step 1.2.7.2.2

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{2}$

Step 1.2.7.2.3

Add $3 π$ and $π$ .

$x = \frac{4 π}{2}$

Step 1.2.7.2.4

Cancel the common factor of $4$ and $2$ .

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Step 1.2.7.2.4.1

Factor $2$ out of $4 π$ .

$x = \frac{2 (2 π)}{2}$

Step 1.2.7.2.4.2

Cancel the common factors.

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Step 1.2.7.2.4.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (2 π)}{2 (1)}$

Step 1.2.7.2.4.2.2

Cancel the common factor.

$x = \frac{2 (2 π)}{2 \cdot 1}$

Step 1.2.7.2.4.2.3

Rewrite the expression.

$x = \frac{2 π}{1}$

Step 1.2.7.2.4.2.4

Divide $2 π$ by $1$ .

$x = 2 π$

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.8.3

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

$\frac{2 π}{1}$

Step 1.2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.9

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

$x = π + 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 1.2.10

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = - 4 \cos ((0) - \frac{π}{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 4 \cos (0 - \frac{π}{2})$

Step 2.2.2

Remove parentheses.

$y = - 4 \cos ((0) - \frac{π}{2})$

Step 2.2.3

Simplify $- 4 \cos ((0) - \frac{π}{2})$ .

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Step 2.2.3.1

Subtract $\frac{π}{2}$ from $0$ .

$y = - 4 \cos (- \frac{π}{2})$

Step 2.2.3.2

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$y = - 4 \cos (\frac{3 π}{2})$

Step 2.2.3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$y = - 4 \cos (\frac{π}{2})$

Step 2.2.3.4

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

$y = - 4 \cdot 0$

Step 2.2.3.5

Multiply $- 4$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(π n, 0)$ , for any integer $n$

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

Step 4