Trigonometry Examples

$2 \sin (x) + 3 = 2.0816$

Step 1

Move all terms not containing $\sin (x)$ to the right side of the equation.

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Step 1.1

Subtract $3$ from both sides of the equation.

$2 \sin (x) = 2.0816 - 3$

Step 1.2

Subtract $3$ from $2.0816$ .

$2 \sin (x) = - 0.9184$

Step 2

Divide each term in $2 \sin (x) = - 0.9184$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \sin (x) = - 0.9184$ by $2$ .

$\frac{2 \sin (x)}{2} = \frac{- 0.9184}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{- 0.9184}{2}$

Step 2.2.1.2

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

$\sin (x) = \frac{- 0.9184}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 0.9184$ by $2$ .

$\sin (x) = - 0.4592$

Step 3

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

$x = \arcsin (- 0.4592)$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (- 0.4592)$ .

$x = - 0.47709442$

Step 5

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.

$x = 2 (3.14159265) + 0.47709442 + 3.14159265$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 (3.14159265) + 0.47709442 + 3.14159265$ .

$x = 2 (3.14159265) + 0.47709442 + 3.14159265 - 2 π$

Step 6.2

The resulting angle of $3.61868707$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.47709442 + 3.14159265$ .

$x = 3.61868707$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

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Step 8.1

Add $2 π$ to $- 0.47709442$ to find the positive angle.

$- 0.47709442 + 2 π$

Step 8.2

Subtract $0.47709442$ from $2 π$ .

$5.80609088$

Step 8.3

List the new angles.

$x = 5.80609088$

Step 9

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

$x = 3.61868707 + 2 π n, 5.80609088 + 2 π n$ , for any integer $n$