Trigonometry Examples

$\sin (3 x - 2 π)$

Step 1

Apply the sum of angles identity.

$\sin (3 x) \cos (- 2 π) + \cos (3 x) \sin (- 2 π)$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the sine triple-angle identity.

$(- 4 \sin^{3} (x) + 3 \sin (x)) \cos (- 2 π) + \cos (3 x) \sin (- 2 π)$

Step 2.1.2

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$(- 4 \sin^{3} (x) + 3 \sin (x)) \cos (0) + \cos (3 x) \sin (- 2 π)$

Step 2.1.3

The exact value of $\cos (0)$ is $1$ .

$(- 4 \sin^{3} (x) + 3 \sin (x)) \cdot 1 + \cos (3 x) \sin (- 2 π)$

Step 2.1.4

Multiply $- 4 \sin^{3} (x) + 3 \sin (x)$ by $1$ .

$- 4 \sin^{3} (x) + 3 \sin (x) + \cos (3 x) \sin (- 2 π)$

Step 2.1.5

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

$- 4 \sin^{3} (x) + 3 \sin (x) + (4 \cos^{3} (x) - 3 \cos (x)) \sin (- 2 π)$

Step 2.1.6

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 2.1.7

The exact value of $\sin (0)$ is $0$ .

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

Step 2.1.8

Multiply $4 \cos^{3} (x) - 3 \cos (x)$ by $0$ .

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

Step 2.2

Add $- 4 \sin^{3} (x) + 3 \sin (x)$ and $0$ .

$- 4 \sin^{3} (x) + 3 \sin (x)$