Trigonometry Examples

$y = - 3 \sin (x + \frac{π}{2})$

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 $x$ .

$0 = - 3 \sin (x + \frac{π}{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 3 \sin (x + \frac{π}{2}) = 0$ .

$- 3 \sin (x + \frac{π}{2}) = 0$

Step 1.2.2

Divide each term in $- 3 \sin (x + \frac{π}{2}) = 0$ by $- 3$ and simplify.

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Step 1.2.2.1

Divide each term in $- 3 \sin (x + \frac{π}{2}) = 0$ by $- 3$ .

$\frac{- 3 \sin (x + \frac{π}{2})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 \sin (x + \frac{π}{2})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2.1.2

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

$\sin (x + \frac{π}{2}) = \frac{0}{- 3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 3$ .

$\sin (x + \frac{π}{2}) = 0$

Step 1.2.3

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

$x + \frac{π}{2} = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$x + \frac{π}{2} = 0$

Step 1.2.5

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = - \frac{π}{2}$

Step 1.2.6

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.

$x + \frac{π}{2} = π - 0$

Step 1.2.7

Solve for $x$ .

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Step 1.2.7.1

Subtract $0$ from $π$ .

$x + \frac{π}{2} = π$

Step 1.2.7.2

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

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Step 1.2.7.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$x = π - \frac{π}{2}$

Step 1.2.7.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.7.2.3

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.7.2.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 1.2.7.2.5

Simplify the numerator.

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Step 1.2.7.2.5.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 1.2.7.2.5.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.8.3

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

$\frac{2 π}{1}$

Step 1.2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.9

Add $2 π$ to every negative angle to get positive angles.

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Step 1.2.9.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 1.2.9.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.9.3

Combine fractions.

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Step 1.2.9.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 1.2.9.4

Simplify the numerator.

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Step 1.2.9.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 1.2.9.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 1.2.9.5

List the new angles.

$x = \frac{3 π}{2}$

Step 1.2.10

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 1.2.11

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

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

$y = - 3 \sin ((0) + \frac{π}{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 3 \sin (0 + \frac{π}{2})$

Step 2.2.2

Remove parentheses.

$y = - 3 \sin ((0) + \frac{π}{2})$

Step 2.2.3

Simplify $- 3 \sin ((0) + \frac{π}{2})$ .

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Step 2.2.3.1

Add $0$ and $\frac{π}{2}$ .

$y = - 3 \sin (\frac{π}{2})$

Step 2.2.3.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$y = - 3 \cdot 1$

Step 2.2.3.3

Multiply $- 3$ by $1$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4