Trigonometry Examples

$f (x) = 4 \sin (2 x - π) - 1$

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 $x$ .

$0 = 4 \sin (2 x - π) - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 \sin (2 x - π) - 1 = 0$ .

$4 \sin (2 x - π) - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$4 \sin (2 x - π) = 1$

Step 1.2.3

Divide each term in $4 \sin (2 x - π) = 1$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 \sin (2 x - π) = 1$ by $4$ .

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

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.

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

Step 1.2.3.2.1.2

Divide $\sin (2 x - π)$ by $1$ .

$\sin (2 x - π) = \frac{1}{4}$

Step 1.2.4

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

$2 x - π = \arcsin (\frac{1}{4})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

Evaluate $\arcsin (\frac{1}{4})$ .

$2 x - π = 0.25268025$

Step 1.2.6

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.6.1

Add $π$ to both sides of the equation.

$2 x = 0.25268025 + π$

Step 1.2.6.2

Replace with decimal approximation.

$2 x = 0.25268025 + 3.14159265$

Step 1.2.6.3

Add $0.25268025$ and $3.14159265$ .

$2 x = 3.3942729$

Step 1.2.7

Divide each term in $2 x = 3.3942729$ by $2$ and simplify.

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Step 1.2.7.1

Divide each term in $2 x = 3.3942729$ by $2$ .

$\frac{2 x}{2} = \frac{3.3942729}{2}$

Step 1.2.7.2

Simplify the left side.

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Step 1.2.7.2.1

Cancel the common factor of $2$ .

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Step 1.2.7.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3.3942729}{2}$

Step 1.2.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.3942729}{2}$

Step 1.2.7.3

Simplify the right side.

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Step 1.2.7.3.1

Divide $3.3942729$ by $2$ .

$x = 1.69713645$

Step 1.2.8

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.

$2 x - π = (3.14159265) - 0.25268025$

Step 1.2.9

Solve for $x$ .

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Step 1.2.9.1

Subtract $0.25268025$ from $3.14159265$ .

$2 x - π = 2.88891239$

Step 1.2.9.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.9.2.1

Add $π$ to both sides of the equation.

$2 x = 2.88891239 + π$

Step 1.2.9.2.2

Replace with decimal approximation.

$2 x = 2.88891239 + 3.14159265$

Step 1.2.9.2.3

Add $2.88891239$ and $3.14159265$ .

$2 x = 6.03050505$

Step 1.2.9.3

Divide each term in $2 x = 6.03050505$ by $2$ and simplify.

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Step 1.2.9.3.1

Divide each term in $2 x = 6.03050505$ by $2$ .

$\frac{2 x}{2} = \frac{6.03050505}{2}$

Step 1.2.9.3.2

Simplify the left side.

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Step 1.2.9.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.9.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6.03050505}{2}$

Step 1.2.9.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6.03050505}{2}$

Step 1.2.9.3.3

Simplify the right side.

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Step 1.2.9.3.3.1

Divide $6.03050505$ by $2$ .

$x = 3.01525252$

Step 1.2.10

Find the period of $\sin (2 x - π)$ .

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Step 1.2.10.1

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

$\frac{2 π}{| b |}$

Step 1.2.10.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.10.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.10.4

Cancel the common factor of $2$ .

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Step 1.2.10.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.10.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.11

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

$x = 1.69713645 + π n, 3.01525252 + π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1.69713645 + π n, 0), (3.01525252 + π n, 0)$ , 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 $y$ .

$y = 4 \sin (2 (0) - π) - 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \sin (2 (0) - π) - 1$

Step 2.2.2

Simplify $4 \sin (2 (0) - π) - 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $0$ .

$y = 4 \sin (0 - π) - 1$

Step 2.2.2.1.2

Subtract $π$ from $0$ .

$y = 4 \sin (- π) - 1$

Step 2.2.2.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$y = 4 \sin (0) - 1$

Step 2.2.2.1.4

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

$y = 4 \cdot 0 - 1$

Step 2.2.2.1.5

Multiply $4$ by $0$ .

$y = 0 - 1$

Step 2.2.2.2

Subtract $1$ from $0$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(1.69713645 + π n, 0), (3.01525252 + π n, 0)$ , for any integer $n$

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

Step 4