Trigonometry Examples

$f (x) = 4 \sin (4 x) - 3$

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 $x$ .

$0 = 4 \sin (4 x) - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 \sin (4 x) - 3 = 0$ .

$4 \sin (4 x) - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$4 \sin (4 x) = 3$

Step 1.2.3

Divide each term in $4 \sin (4 x) = 3$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 \sin (4 x) = 3$ by $4$ .

$\frac{4 \sin (4 x)}{4} = \frac{3}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 \sin (4 x)}{4} = \frac{3}{4}$

Step 1.2.3.2.1.2

Divide $\sin (4 x)$ by $1$ .

$\sin (4 x) = \frac{3}{4}$

Step 1.2.4

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

$4 x = \arcsin (\frac{3}{4})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

Evaluate $\arcsin (\frac{3}{4})$ .

$4 x = 0.84806207$

Step 1.2.6

Divide each term in $4 x = 0.84806207$ by $4$ and simplify.

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Step 1.2.6.1

Divide each term in $4 x = 0.84806207$ by $4$ .

$\frac{4 x}{4} = \frac{0.84806207}{4}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $4$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0.84806207}{4}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.84806207}{4}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Divide $0.84806207$ by $4$ .

$x = 0.21201551$

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.

$4 x = (3.14159265) - 0.84806207$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Subtract $0.84806207$ from $3.14159265$ .

$4 x = 2.29353057$

Step 1.2.8.2

Divide each term in $4 x = 2.29353057$ by $4$ and simplify.

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Step 1.2.8.2.1

Divide each term in $4 x = 2.29353057$ by $4$ .

$\frac{4 x}{4} = \frac{2.29353057}{4}$

Step 1.2.8.2.2

Simplify the left side.

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Step 1.2.8.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.8.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2.29353057}{4}$

Step 1.2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.29353057}{4}$

Step 1.2.8.2.3

Simplify the right side.

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Step 1.2.8.2.3.1

Divide $2.29353057$ by $4$ .

$x = 0.57338264$

Step 1.2.9

Find the period of $\sin (4 x)$ .

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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 $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 1.2.9.4

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

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Step 1.2.9.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 1.2.9.4.2

Cancel the common factors.

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Step 1.2.9.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 1.2.9.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 1.2.9.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 1.2.10

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

$x = 0.21201551 + \frac{π n}{2}, 0.57338264 + \frac{π n}{2}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0.21201551 + \frac{π n}{2}, 0), (0.57338264 + \frac{π n}{2}, 0)$ , 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 $y$ .

$y = 4 \sin (4 (0)) - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \sin (4 (0)) - 3$

Step 2.2.2

Simplify $4 \sin (4 (0)) - 3$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $4$ by $0$ .

$y = 4 \sin (0) - 3$

Step 2.2.2.1.2

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

$y = 4 \cdot 0 - 3$

Step 2.2.2.1.3

Multiply $4$ by $0$ .

$y = 0 - 3$

Step 2.2.2.2

Subtract $3$ from $0$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(0.21201551 + \frac{π n}{2}, 0), (0.57338264 + \frac{π n}{2}, 0)$ , for any integer $n$

y-intercept(s): $(0, - 3)$

Step 4