Trigonometry Examples

sin (4 x) cos (x) - sin (x) cos (4 x) = - \frac{\sqrt{2}}{2}

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

[0, 2 π)

$[0, 2 π)$

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}

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

n

$n$

Step 2

Find the values of

n

$n$ that produce a value within the interval

[0, 2 π)

$[0, 2 π)$ .

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Step 2.1

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 2.1.1

Plug in

0

$0$ for

n

$n$ .

\frac{5 π}{12} + \frac{2 π (0)}{3}

$\frac{5 π}{12} + \frac{2 π (0)}{3}$

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

$0$ and

3

$3$ .

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Step 2.1.2.1.1.1

Factor

3

$3$ out of

2 π (0)

$2 π (0)$ .

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

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

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

$3$ out of

3

$3$ .

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

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

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