Trigonometry Examples

$\log (4) \sin (x) = - 0.5$

Step 1

Divide each term in $\log (4) \sin (x) = - 0.5$ by $\log (4)$ and simplify.

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Step 1.1

Divide each term in $\log (4) \sin (x) = - 0.5$ by $\log (4)$ .

$\frac{\log (4) \sin (x)}{\log (4)} = \frac{- 0.5}{\log (4)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\log (4)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\log (4) \sin (x)}{\log (4)} = \frac{- 0.5}{\log (4)}$

Step 1.2.1.2

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

$\sin (x) = \frac{- 0.5}{\log (4)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{0.5}{\log (4)}$

Step 2

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

$x = \arcsin (- \frac{0.5}{\log (4)})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\arcsin (- \frac{0.5}{\log (4)})$ .

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Step 3.1.1

Log base $10$ of $4$ is approximately $0.60205999$ .

$x = \arcsin (- \frac{0.5}{0.60205999})$

Step 3.1.2

Divide $0.5$ by $0.60205999$ .

$x = \arcsin (- 1 \cdot 0.83048202)$

Step 3.1.3

Multiply $- 1$ by $0.83048202$ .

$x = \arcsin (- 0.83048202)$

Step 3.1.4

Evaluate $\arcsin (- 0.83048202)$ .

$x = - 0.97997244$

Step 4

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.

$x = 2 (3.14159265) + 0.97997244 + 3.14159265$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 (3.14159265) + 0.97997244 + 3.14159265$ .

$x = 2 (3.14159265) + 0.97997244 + 3.14159265 - 2 π$

Step 5.2

The resulting angle of $4.1215651$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.97997244 + 3.14159265$ .

$x = 4.1215651$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

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Step 7.1

Add $2 π$ to $- 0.97997244$ to find the positive angle.

$- 0.97997244 + 2 π$

Step 7.2

Subtract $0.97997244$ from $2 π$ .

$5.30321285$

Step 7.3

List the new angles.

$x = 5.30321285$

Step 8

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

$x = 4.1215651 + 2 π n, 5.30321285 + 2 π n$ , for any integer $n$