Trigonometry Examples

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

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

[0, 2 π]

$[0, 2 π]$

Step 1

Simplify each term.

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Step 1.1

Multiply

\sin (x)

$\sin (x)$ by

\sin (x)

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

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Step 1.1.1

Move

\sin (x)

$\sin (x)$ .

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

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

Step 1.1.2

Multiply

\sin (x)

$\sin (x)$ by

\sin (x)

$\sin (x)$ .

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

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

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

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

Step 1.2

Multiply

\sin^{2} (x)

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

\sin (x)

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

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Step 1.2.1

Move

\sin (x)

$\sin (x)$ .

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

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

Step 1.2.2

Multiply

\sin (x)

$\sin (x)$ by

\sin^{2} (x)

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

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Step 1.2.2.1

Raise

\sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 1.2.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 1.2.3

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 1.3

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

n

$n$

Step 3

Find the values of

n

$n$ that produce a value within the interval

[0, 2 π]

$[0, 2 π]$ .

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Step 3.1

Plug in

0

$0$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π]

$[0, 2 π]$ .

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Step 3.1.1

Plug in

0

$0$ for

n

$n$ .

0.89164272 + π (0)

$0.89164272 + π (0)$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Multiply

π

$π$ by

0

$0$ .

0.89164272 + 0

$0.89164272 + 0$

Step 3.1.2.2

Add

0.89164272

$0.89164272$ and

0

$0$ .

0.89164272

$0.89164272$

0.89164272

$0.89164272$

Step 3.1.3

The interval

[0, 2 π]

$[0, 2 π]$ contains

0.89164272

$0.89164272$ .

x = 0.89164272

$x = 0.89164272$

x = 0.89164272

$x = 0.89164272$

Step 3.2

Plug in

1

$1$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π]

$[0, 2 π]$ .

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Step 3.2.1

Plug in

1

$1$ for

n

$n$ .

0.89164272 + π (1)

$0.89164272 + π (1)$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply

π

$π$ by

1

$1$ .

0.89164272 + π

$0.89164272 + π$

Step 3.2.2.2

Replace with decimal approximation.

0.89164272 + 3.14159265

$0.89164272 + 3.14159265$

Step 3.2.2.3

Add

0.89164272

$0.89164272$ and

3.14159265

$3.14159265$ .

4.03323537

$4.03323537$

4.03323537

$4.03323537$

Step 3.2.3

The interval

[0, 2 π]

$[0, 2 π]$ contains

4.03323537

$4.03323537$ .

x = 0.89164272, 4.03323537

$x = 0.89164272, 4.03323537$

x = 0.89164272, 4.03323537

$x = 0.89164272, 4.03323537$

x = 0.89164272, 4.03323537

$x = 0.89164272, 4.03323537$

Step 4