Trigonometry Examples

$y = 4 \sin (\frac{1}{2} x)$

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 $x$ .

$0 = 4 \sin (\frac{1}{2} x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 \sin (\frac{1}{2} x) = 0$ .

$4 \sin (\frac{1}{2} x) = 0$

Step 1.2.2

Divide each term in $4 \sin (\frac{1}{2} x) = 0$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 \sin (\frac{1}{2} x) = 0$ by $4$ .

$\frac{4 \sin (\frac{1}{2} x)}{4} = \frac{0}{4}$

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.

$\frac{4 \sin (\frac{1}{2} x)}{4} = \frac{0}{4}$

Step 1.2.2.2.1.2

Divide $\sin (\frac{1}{2} x)$ by $1$ .

$\sin (\frac{1}{2} x) = \frac{0}{4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $4$ .

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the left side.

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Step 1.2.4.1

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

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

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{x}{2} = 0$

Step 1.2.6

Set the numerator equal to zero.

$x = 0$

Step 1.2.7

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.

$\frac{x}{2} = π - 0$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - 0)$

Step 1.2.8.2

Simplify both sides of the equation.

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Step 1.2.8.2.1

Simplify the left side.

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Step 1.2.8.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.8.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - 0)$

Step 1.2.8.2.1.1.2

Rewrite the expression.

$x = 2 (π - 0)$

Step 1.2.8.2.2

Simplify the right side.

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Step 1.2.8.2.2.1

Subtract $0$ from $π$ .

$x = 2 π$

Step 1.2.9

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

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Step 1.2.9.1

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

$\frac{2 π}{| b |}$

Step 1.2.9.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 1.2.9.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 1.2.9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 1.2.9.5

Multiply $2$ by $2$ .

$4 π$

Step 1.2.10

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

$x = 4 π n, 2 π + 4 π n$ , for any integer $n$

Step 1.2.11

Consolidate the answers.

$x = 2 π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2 π n, 0)$ , 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 $y$ .

$y = 4 \sin (\frac{1}{2} \cdot (0))$

Step 2.2

Solve the equation.

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Step 2.2.1

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

$y = 4 \sin (\frac{1}{2} \cdot 0)$

Step 2.2.2

Remove parentheses.

$y = 4 \sin (\frac{1}{2} \cdot (0))$

Step 2.2.3

Simplify $4 \sin (\frac{1}{2} \cdot (0))$ .

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Step 2.2.3.1

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

$y = 4 \sin (0)$

Step 2.2.3.2

The exact value of $\sin (0)$ is $0$ .

$y = 4 \cdot 0$

Step 2.2.3.3

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): $(2 π n, 0)$ , for any integer $n$

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

Step 4