Trigonometry Examples

f (x) = 2 sin (x)

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = 2 sin (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

$2 \sin (x) = 0$ .

2 sin (x) = 0

$2 \sin (x) = 0$

Step 1.2.2

Divide each term in

2 sin (x) = 0

$2 \sin (x) = 0$ by

2

$2$ and simplify.

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Step 1.2.2.1

Divide each term in

2 sin (x) = 0

$2 \sin (x) = 0$ by

2

$2$ .

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

$2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

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

Step 1.2.2.2.1.2

Divide

sin (x)

$\sin (x)$ by

1

$1$ .

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

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

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

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

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

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide

0

$0$ by

2

$2$ .

sin (x) = 0

$\sin (x) = 0$

sin (x) = 0

$\sin (x) = 0$

sin (x) = 0

$\sin (x) = 0$

Step 1.2.3

Take the inverse sine of both sides of the equation to extract

x

$x$ from inside the sine.

x = arcsin (0)

$x = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

The exact value of

arcsin (0)

$\arcsin (0)$ is

0

$0$ .

x = 0

$x = 0$

x = 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

$π$ to find the solution in the second quadrant.

x = π - 0

$x = π - 0$

Step 1.2.6

Subtract

0

$0$ from

π

$π$ .

x = π

$x = π$

Step 1.2.7

Find the period of

sin (x)

$\sin (x)$ .

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Step 1.2.7.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 1.2.7.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 1.2.7.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 1.2.7.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 1.2.8

The period of the

sin (x)

$\sin (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = 2 π n, π + 2 π n

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

n

$n$

Step 1.2.9

Consolidate the answers.

x = π n

$x = π n$ , for any integer

n

$n$

x = π n

$x = π n$ , for any integer

n

$n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s):

(π n, 0)

$(π n, 0)$ , for any integer

n

$n$

x-intercept(s):

(π n, 0)

$(π n, 0)$ , for any integer

n

$n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y = 2 sin (0)

$y = 2 \sin (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

y = 2 sin (0)

$y = 2 \sin (0)$

Step 2.2.2

Simplify

2 sin (0)

$2 \sin (0)$ .

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Step 2.2.2.1

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

y = 2 \cdot 0

$y = 2 \cdot 0$

Step 2.2.2.2

Multiply

2

$2$ by

0

$0$ .

y = 0

$y = 0$

y = 0

$y = 0$

y = 0

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

(0, 0)

$(0, 0)$

y-intercept(s):

(0, 0)

$(0, 0)$

Step 3

List the intersections.

x-intercept(s):

(π n, 0)

$(π n, 0)$ , for any integer

n

$n$

y-intercept(s):

(0, 0)

$(0, 0)$

Step 4