Trigonometry Examples

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

Step 1

Write $\cos (\frac{1}{2} x)$ as an equation.

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

Step 2

Find the x-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the left side.

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Step 2.2.3.1

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

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

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

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

$\frac{x}{2} = \frac{π}{2}$

Step 2.2.5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

Step 2.2.6

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.

$\frac{x}{2} = 2 π - \frac{π}{2}$

Step 2.2.7

Solve for $x$ .

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Step 2.2.7.1

Multiply both sides of the equation by $2$ .

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

Step 2.2.7.2

Simplify both sides of the equation.

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Step 2.2.7.2.1

Simplify the left side.

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Step 2.2.7.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.7.2.1.1.1

Cancel the common factor.

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

Step 2.2.7.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - \frac{π}{2})$

Step 2.2.7.2.2

Simplify the right side.

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Step 2.2.7.2.2.1

Simplify $2 (2 π - \frac{π}{2})$ .

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Step 2.2.7.2.2.1.1

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

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

Step 2.2.7.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.2.7.2.2.1.3

Combine the numerators over the common denominator.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 2.2.7.2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.7.2.2.1.4.1

Cancel the common factor.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 2.2.7.2.2.1.4.2

Rewrite the expression.

$x = 2 π \cdot 2 - π$

Step 2.2.7.2.2.1.5

Multiply $2$ by $2$ .

$x = 4 π - π$

Step 2.2.7.2.2.1.6

Subtract $π$ from $4 π$ .

$x = 3 π$

Step 2.2.8

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

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Step 2.2.8.1

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

$\frac{2 π}{| b |}$

Step 2.2.8.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 2.2.8.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 2.2.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 2.2.8.5

Multiply $2$ by $2$ .

$4 π$

Step 2.2.9

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

$x = π + 4 π n, 3 π + 4 π n$ , for any integer $n$

Step 2.2.10

Consolidate the answers.

$x = π + 2 π n$ , for any integer $n$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(π + 2 π n, 0)$ , for any integer $n$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = \cos (\frac{1}{2} \cdot (0))$

Step 3.2

Solve the equation.

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Step 3.2.1

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

$y = \cos (\frac{1}{2} \cdot 0)$

Step 3.2.2

Remove parentheses.

$y = \cos (\frac{1}{2} \cdot (0))$

Step 3.2.3

Simplify $\cos (\frac{1}{2} \cdot (0))$ .

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Step 3.2.3.1

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

$y = \cos (0)$

Step 3.2.3.2

The exact value of $\cos (0)$ is $1$ .

$y = 1$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 4

List the intersections.

x-intercept(s): $(π + 2 π n, 0)$ , for any integer $n$

y-intercept(s): $(0, 1)$

Step 5