Trigonometry Examples

$\sin (b x - d) = 0$

Step 1

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

$b x - d = \arcsin (0)$

Step 2

Simplify the right side.

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Step 2.1

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

$b x - d = 0$

Step 3

Add $d$ to both sides of the equation.

$b x = d$

Step 4

Divide each term in $b x = d$ by $b$ and simplify.

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Step 4.1

Divide each term in $b x = d$ by $b$ .

$\frac{b x}{b} = \frac{d}{b}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $b$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{b x}{b} = \frac{d}{b}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{d}{b}$

Step 5

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.

$b x - d = π - 0$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $0$ from $π$ .

$b x - d = π$

Step 6.2

Add $d$ to both sides of the equation.

$b x = π + d$

Step 6.3

Divide each term in $b x = π + d$ by $b$ and simplify.

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Step 6.3.1

Divide each term in $b x = π + d$ by $b$ .

$\frac{b x}{b} = \frac{π}{b} + \frac{d}{b}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $b$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{b x}{b} = \frac{π}{b} + \frac{d}{b}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{b} + \frac{d}{b}$