Precalculus Examples

$\cos (X + \frac{5 π}{2}) = - \sin (X)$

Step 1

Start on the left side.

$\cos (X + \frac{5 π}{2})$

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\cos (X) \cos (\frac{5 π}{2}) - \sin (X) \sin (\frac{5 π}{2})$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$\cos (X) \cos (\frac{π}{2}) - \sin (X) \sin (\frac{5 π}{2})$

Step 3.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\cos (X) \cdot 0 - \sin (X) \sin (\frac{5 π}{2})$

Step 3.1.3

Multiply $\cos (X)$ by $0$ .

$0 - \sin (X) \sin (\frac{5 π}{2})$

Step 3.1.4

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

$0 - \sin (X) \sin (\frac{π}{2})$

Step 3.1.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 - \sin (X) \cdot 1$

Step 3.1.6

Multiply $- 1$ by $1$ .

$0 - \sin (X)$

Step 3.2

Subtract $\sin (X)$ from $0$ .

$- \sin (X)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos (X + \frac{5 π}{2}) = - \sin (X)$ is an identity