Trigonometry Examples

$f (x) = 2 \sin (x)$

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 $x$ .

$0 = 2 \sin (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 \sin (x) = 0$ .

$2 \sin (x) = 0$

Step 1.2.2

Divide each term in $2 \sin (x) = 0$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 \sin (x) = 0$ by $2$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $2$ .

$\sin (x) = 0$

Step 1.2.3

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

$x = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$x = 0$

Step 1.2.5

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.

$x = π - 0$

Step 1.2.6

Subtract $0$ from $π$ .

$x = π$

Step 1.2.7

Find the period of $\sin (x)$ .

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Step 1.2.7.1

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

$\frac{2 π}{| b |}$

Step 1.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.7.3

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

$\frac{2 π}{1}$

Step 1.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.8

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

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

Step 1.2.9

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 $y$ .

$y = 2 \sin (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \sin (0)$

Step 2.2.2

Simplify $2 \sin (0)$ .

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Step 2.2.2.1

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

$y = 2 \cdot 0$

Step 2.2.2.2

Multiply $2$ 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