Trigonometry Examples

$2 (2 \sin (x) \sin (x)) \sin (x) - 3 \cos (x) = 0$ , $[0, 2 π]$

Step 1

Simplify each term.

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Step 1.1

Multiply $\sin (x)$ by $\sin (x)$ by adding the exponents.

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Step 1.1.1

Move $\sin (x)$ .

$2 (2 (\sin (x) \sin (x))) \sin (x) - 3 \cos (x) = 0$

Step 1.1.2

Multiply $\sin (x)$ by $\sin (x)$ .

$2 (2 \sin^{2} (x)) \sin (x) - 3 \cos (x) = 0$

Step 1.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 1.2.1

Move $\sin (x)$ .

$2 (2 (\sin (x) \sin^{2} (x))) - 3 \cos (x) = 0$

Step 1.2.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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Step 1.2.2.1

Raise $\sin (x)$ to the power of $1$ .

$2 (2 (\sin^{1} (x) \sin^{2} (x))) - 3 \cos (x) = 0$

Step 1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (2 {\sin (x)}^{1 + 2}) - 3 \cos (x) = 0$

Step 1.2.3

Add $1$ and $2$ .

$2 (2 \sin^{3} (x)) - 3 \cos (x) = 0$

Step 1.3

Multiply $2$ by $2$ .

$4 \sin^{3} (x) - 3 \cos (x) = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.89164272 + π n$ , for any integer $n$

Step 3

Find the values of $n$ that produce a value within the interval $[0, 2 π]$ .

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Step 3.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π]$ .

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Step 3.1.1

Plug in $0$ for $n$ .

$0.89164272 + π (0)$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Multiply $π$ by $0$ .

$0.89164272 + 0$

Step 3.1.2.2

Add $0.89164272$ and $0$ .

$0.89164272$

Step 3.1.3

The interval $[0, 2 π]$ contains $0.89164272$ .

$x = 0.89164272$

Step 3.2

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π]$ .

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Step 3.2.1

Plug in $1$ for $n$ .

$0.89164272 + π (1)$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply $π$ by $1$ .

$0.89164272 + π$

Step 3.2.2.2

Replace with decimal approximation.

$0.89164272 + 3.14159265$

Step 3.2.2.3

Add $0.89164272$ and $3.14159265$ .

$4.03323537$

Step 3.2.3

The interval $[0, 2 π]$ contains $4.03323537$ .

$x = 0.89164272, 4.03323537$

Step 4