Trigonometry Examples

$\sin (x + 2 k π)$

Step 1

Apply the sum of angles identity.

$\sin (x) \cos (2 k π) + \cos (x) \sin (2 k π)$

Step 2

Simplify each term.

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Step 2.1

Add parentheses.

$\sin (x) \cos (2 (k π)) + \cos (x) \sin (2 k π)$

Step 2.2

Use the double-angle identity to transform $\cos (2 x)$ to $\cos^{2} (x) - \sin^{2} (x)$ .

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Add parentheses.

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

Step 2.6

Apply the sine double-angle identity.

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

Step 2.7

Rewrite using the commutative property of multiplication.

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