Trigonometry Examples

$\sin (2 x + 1.5) = - 0.3$

Step 1

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

$2 x + 1.5 = \arcsin (- 0.3)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (- 0.3)$ .

$2 x + 1.5 = - 0.30469265$

Step 3

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

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Step 3.1

Subtract $1.5$ from both sides of the equation.

$2 x = - 0.30469265 - 1.5$

Step 3.2

Subtract $1.5$ from $- 0.30469265$ .

$2 x = - 1.80469265$

Step 4

Divide each term in $2 x = - 1.80469265$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - 1.80469265$ by $2$ .

$\frac{2 x}{2} = \frac{- 1.80469265}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1.80469265}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1.80469265}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 1.80469265$ by $2$ .

$x = - 0.90234632$

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.

$2 x + 1.5 = 2 (3.14159265) + 0.30469265 + 3.14159265$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 (3.14159265) + 0.30469265 + 3.14159265$ .

$2 x + 1.5 = 2 (3.14159265) + 0.30469265 + 3.14159265 - 2 π$

Step 6.2

The resulting angle of $3.4462853$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.30469265 + 3.14159265$ .

$2 x + 1.5 = 3.4462853$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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Step 6.3.1.1

Subtract $1.5$ from both sides of the equation.

$2 x = 3.4462853 - 1.5$

Step 6.3.1.2

Subtract $1.5$ from $3.4462853$ .

$2 x = 1.9462853$

Step 6.3.2

Divide each term in $2 x = 1.9462853$ by $2$ and simplify.

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Step 6.3.2.1

Divide each term in $2 x = 1.9462853$ by $2$ .

$\frac{2 x}{2} = \frac{1.9462853}{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.9462853}{2}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.9462853}{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $1.9462853$ by $2$ .

$x = 0.97314265$

Step 7

Find the period of $\sin (2 x + 1.5)$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

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Step 8.1

Add $π$ to $- 0.90234632$ to find the positive angle.

$- 0.90234632 + π$

Step 8.2

Replace with decimal approximation.

$3.14159265 - 0.90234632$

Step 8.3

Subtract $0.90234632$ from $3.14159265$ .

$2.23924632$

Step 8.4

List the new angles.

$x = 2.23924632$

Step 9

The period of the $\sin (2 x + 1.5)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 0.97314265 + π n, 2.23924632 + π n$ , for any integer $n$