Trigonometry Examples

$\sin (4 x) \cos (x) - \sin (x) \cos (4 x) = - \frac{\sqrt{2}}{2}$ , $[0, 2 π)$

Step 1

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

$x = \frac{5 π}{12} + \frac{2 π n}{3}, \frac{7 π}{12} + \frac{2 π n}{3}$ , for any integer $n$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Plug in $0$ for $n$ .

$\frac{5 π}{12} + \frac{2 π (0)}{3}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Cancel the common factor of $0$ and $3$ .

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Step 2.1.2.1.1.1

Factor $3$ out of $2 π (0)$ .

$\frac{5 π}{12} + \frac{3 (2 π \cdot (0))}{3}$

Step 2.1.2.1.1.2

Cancel the common factors.

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Step 2.1.2.1.1.2.1

Factor $3$ out of $3$ .

$\frac{5 π}{12} + \frac{3 (2 π \cdot (0))}{3 (1)}$

Step 2.1.2.1.1.2.2

Cancel the common factor.

$\frac{5 π}{12} + \frac{3 (2 π \cdot (0))}{3 \cdot 1}$

Step 2.1.2.1.1.2.3

Rewrite the expression.

$\frac{5 π}{12} + \frac{2 π \cdot (0)}{1}$

Step 2.1.2.1.1.2.4

Divide $2 π \cdot (0)$ by $1$ .

$\frac{5 π}{12} + 2 π \cdot (0)$

Step 2.1.2.1.2

Multiply $2 π (0)$ .

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Step 2.1.2.1.2.1

Multiply $0$ by $2$ .

$\frac{5 π}{12} + 0 π$

Step 2.1.2.1.2.2

Multiply $0$ by $π$ .

$\frac{5 π}{12} + 0$

Step 2.1.2.2

Add $\frac{5 π}{12}$ and $0$ .

$\frac{5 π}{12}$

Step 2.1.3

The interval $[0, 2 π)$ contains $\frac{5 π}{12}$ .

$x = \frac{5 π}{12}$

Step 2.2

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

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Step 2.2.1

Plug in $0$ for $n$ .

$\frac{7 π}{12} + \frac{2 π (0)}{3}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $0$ and $3$ .

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Step 2.2.2.1.1.1

Factor $3$ out of $2 π (0)$ .

$\frac{7 π}{12} + \frac{3 (2 π \cdot (0))}{3}$

Step 2.2.2.1.1.2

Cancel the common factors.

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Step 2.2.2.1.1.2.1

Factor $3$ out of $3$ .

$\frac{7 π}{12} + \frac{3 (2 π \cdot (0))}{3 (1)}$

Step 2.2.2.1.1.2.2

Cancel the common factor.

$\frac{7 π}{12} + \frac{3 (2 π \cdot (0))}{3 \cdot 1}$

Step 2.2.2.1.1.2.3

Rewrite the expression.

$\frac{7 π}{12} + \frac{2 π \cdot (0)}{1}$

Step 2.2.2.1.1.2.4

Divide $2 π \cdot (0)$ by $1$ .

$\frac{7 π}{12} + 2 π \cdot (0)$

Step 2.2.2.1.2

Multiply $2 π (0)$ .

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Step 2.2.2.1.2.1

Multiply $0$ by $2$ .

$\frac{7 π}{12} + 0 π$

Step 2.2.2.1.2.2

Multiply $0$ by $π$ .

$\frac{7 π}{12} + 0$

Step 2.2.2.2

Add $\frac{7 π}{12}$ and $0$ .

$\frac{7 π}{12}$

Step 2.2.3

The interval $[0, 2 π)$ contains $\frac{7 π}{12}$ .

$x = \frac{5 π}{12}, \frac{7 π}{12}$

Step 2.3

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

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Step 2.3.1

Plug in $1$ for $n$ .

$\frac{5 π}{12} + \frac{2 π (1)}{3}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2$ by $1$ .

$\frac{5 π}{12} + \frac{2 π}{3}$

Step 2.3.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5 π}{12} + \frac{2 π}{3} \cdot \frac{4}{4}$

Step 2.3.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{2 π}{3}$ by $\frac{4}{4}$ .

$\frac{5 π}{12} + \frac{2 π \cdot 4}{3 \cdot 4}$

Step 2.3.2.3.2

Multiply $3$ by $4$ .

$\frac{5 π}{12} + \frac{2 π \cdot 4}{12}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$\frac{5 π + 2 π \cdot 4}{12}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $4$ by $2$ .

$\frac{5 π + 8 π}{12}$

Step 2.3.2.5.2

Add $5 π$ and $8 π$ .

$\frac{13 π}{12}$

Step 2.3.3

The interval $[0, 2 π)$ contains $\frac{13 π}{12}$ .

$x = \frac{5 π}{12}, \frac{7 π}{12}, \frac{13 π}{12}$

Step 2.4

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

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Step 2.4.1

Plug in $1$ for $n$ .

$\frac{7 π}{12} + \frac{2 π (1)}{3}$

Step 2.4.2

Simplify.

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Step 2.4.2.1

Multiply $2$ by $1$ .

$\frac{7 π}{12} + \frac{2 π}{3}$

Step 2.4.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7 π}{12} + \frac{2 π}{3} \cdot \frac{4}{4}$

Step 2.4.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.2.3.1

Multiply $\frac{2 π}{3}$ by $\frac{4}{4}$ .

$\frac{7 π}{12} + \frac{2 π \cdot 4}{3 \cdot 4}$

Step 2.4.2.3.2

Multiply $3$ by $4$ .

$\frac{7 π}{12} + \frac{2 π \cdot 4}{12}$

Step 2.4.2.4

Combine the numerators over the common denominator.

$\frac{7 π + 2 π \cdot 4}{12}$

Step 2.4.2.5

Simplify the numerator.

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Step 2.4.2.5.1

Multiply $4$ by $2$ .

$\frac{7 π + 8 π}{12}$

Step 2.4.2.5.2

Add $7 π$ and $8 π$ .

$\frac{15 π}{12}$

Step 2.4.2.6

Cancel the common factor of $15$ and $12$ .

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Step 2.4.2.6.1

Factor $3$ out of $15 π$ .

$\frac{3 (5 π)}{12}$

Step 2.4.2.6.2

Cancel the common factors.

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Step 2.4.2.6.2.1

Factor $3$ out of $12$ .

$\frac{3 (5 π)}{3 (4)}$

Step 2.4.2.6.2.2

Cancel the common factor.

$\frac{3 (5 π)}{3 \cdot 4}$

Step 2.4.2.6.2.3

Rewrite the expression.

$\frac{5 π}{4}$

Step 2.4.3

The interval $[0, 2 π)$ contains $\frac{5 π}{4}$ .

$x = \frac{5 π}{12}, \frac{7 π}{12}, \frac{13 π}{12}, \frac{5 π}{4}$

Step 2.5

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

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Step 2.5.1

Plug in $2$ for $n$ .

$\frac{5 π}{12} + \frac{2 π (2)}{3}$

Step 2.5.2

Simplify.

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Step 2.5.2.1

Multiply $2$ by $2$ .

$\frac{5 π}{12} + \frac{4 π}{3}$

Step 2.5.2.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5 π}{12} + \frac{4 π}{3} \cdot \frac{4}{4}$

Step 2.5.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.2.3.1

Multiply $\frac{4 π}{3}$ by $\frac{4}{4}$ .

$\frac{5 π}{12} + \frac{4 π \cdot 4}{3 \cdot 4}$

Step 2.5.2.3.2

Multiply $3$ by $4$ .

$\frac{5 π}{12} + \frac{4 π \cdot 4}{12}$

Step 2.5.2.4

Combine the numerators over the common denominator.

$\frac{5 π + 4 π \cdot 4}{12}$

Step 2.5.2.5

Simplify the numerator.

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Step 2.5.2.5.1

Multiply $4$ by $4$ .

$\frac{5 π + 16 π}{12}$

Step 2.5.2.5.2

Add $5 π$ and $16 π$ .

$\frac{21 π}{12}$

Step 2.5.2.6

Cancel the common factor of $21$ and $12$ .

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Step 2.5.2.6.1

Factor $3$ out of $21 π$ .

$\frac{3 (7 π)}{12}$

Step 2.5.2.6.2

Cancel the common factors.

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Step 2.5.2.6.2.1

Factor $3$ out of $12$ .

$\frac{3 (7 π)}{3 (4)}$

Step 2.5.2.6.2.2

Cancel the common factor.

$\frac{3 (7 π)}{3 \cdot 4}$

Step 2.5.2.6.2.3

Rewrite the expression.

$\frac{7 π}{4}$

Step 2.5.3

The interval $[0, 2 π)$ contains $\frac{7 π}{4}$ .

$x = \frac{5 π}{12}, \frac{7 π}{12}, \frac{13 π}{12}, \frac{5 π}{4}, \frac{7 π}{4}$

Step 2.6

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

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Step 2.6.1

Plug in $2$ for $n$ .

$\frac{7 π}{12} + \frac{2 π (2)}{3}$

Step 2.6.2

Simplify.

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Step 2.6.2.1

Multiply $2$ by $2$ .

$\frac{7 π}{12} + \frac{4 π}{3}$

Step 2.6.2.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7 π}{12} + \frac{4 π}{3} \cdot \frac{4}{4}$

Step 2.6.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.2.3.1

Multiply $\frac{4 π}{3}$ by $\frac{4}{4}$ .

$\frac{7 π}{12} + \frac{4 π \cdot 4}{3 \cdot 4}$

Step 2.6.2.3.2

Multiply $3$ by $4$ .

$\frac{7 π}{12} + \frac{4 π \cdot 4}{12}$

Step 2.6.2.4

Combine the numerators over the common denominator.

$\frac{7 π + 4 π \cdot 4}{12}$

Step 2.6.2.5

Simplify the numerator.

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Step 2.6.2.5.1

Multiply $4$ by $4$ .

$\frac{7 π + 16 π}{12}$

Step 2.6.2.5.2

Add $7 π$ and $16 π$ .

$\frac{23 π}{12}$

Step 2.6.3

The interval $[0, 2 π)$ contains $\frac{23 π}{12}$ .

$x = \frac{5 π}{12}, \frac{7 π}{12}, \frac{13 π}{12}, \frac{5 π}{4}, \frac{7 π}{4}, \frac{23 π}{12}$

Step 3