Trigonometry Examples

$f (x) = - 2 \sin (x) - 1$

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 $x$ .

$0 = - 2 \sin (x) - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$- 2 \sin (x) - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$- 2 \sin (x) = 1$

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

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

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Move the negative in front of the fraction.

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

Step 1.2.4

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

$x = \arcsin (- \frac{1}{2})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 1.2.6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Step 1.2.7

Simplify the expression to find the second solution.

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Step 1.2.7.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 1.2.7.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 1.2.8

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

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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 $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 1.2.9.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 1.2.9.3

Combine fractions.

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Step 1.2.9.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 1.2.9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 1.2.9.4

Simplify the numerator.

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Step 1.2.9.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 1.2.9.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 1.2.9.5

List the new angles.

$x = \frac{11 π}{6}$

Step 1.2.10

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{7 π}{6} + 2 π n, 0), (\frac{11 π}{6} + 2 π n, 0)$ , 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 $y$ .

$y = - 2 \sin (0) - 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 2 \sin (0) - 1$

Step 2.2.2

Simplify $- 2 \sin (0) - 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$y = - 2 \cdot 0 - 1$

Step 2.2.2.1.2

Multiply $- 2$ 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): $(\frac{7 π}{6} + 2 π n, 0), (\frac{11 π}{6} + 2 π n, 0)$ , for any integer $n$

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

Step 4