Trigonometry Examples

$\cos (x - \frac{5 π}{4}) = - \frac{\sqrt{2}}{2} \cdot (\cos (x) + \sin (x))$

Step 1

Start on the left side.

$\cos (x - \frac{5 π}{4})$

Step 2

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

$\cos (x) \cos (\frac{5 π}{4}) + \sin (x) \sin (\frac{5 π}{4})$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\cos (x) (- \cos (\frac{π}{4})) + \sin (x) \sin (\frac{5 π}{4})$

Step 3.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\cos (x) (- \frac{\sqrt{2}}{2}) + \sin (x) \sin (\frac{5 π}{4})$

Step 3.3

Combine $\cos (x)$ and $\frac{\sqrt{2}}{2}$ .

$- \frac{\cos (x) \sqrt{2}}{2} + \sin (x) \sin (\frac{5 π}{4})$

Step 3.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$- \frac{\cos (x) \sqrt{2}}{2} + \sin (x) (- \sin (\frac{π}{4}))$

Step 3.5

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\cos (x) \sqrt{2}}{2} + \sin (x) (- \frac{\sqrt{2}}{2})$

Step 3.6

Combine $\sin (x)$ and $\frac{\sqrt{2}}{2}$ .

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

Step 4

Reorder factors in $- \frac{\cos (x) \sqrt{2}}{2} - \frac{\sin (x) \sqrt{2}}{2}$ .

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

Step 5

Now consider the right side of the equation.

$- \frac{\sqrt{2}}{2} \cdot (\cos (x) + \sin (x))$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Combine $\cos (x)$ and $\frac{\sqrt{2}}{2}$ .

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

Step 6.3

Combine $\sin (x)$ and $\frac{\sqrt{2}}{2}$ .

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

Step 6.4

Reorder factors in $- \frac{\cos (x) \sqrt{2}}{2} - \frac{\sin (x) \sqrt{2}}{2}$ .

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

Step 7

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

$\cos (x - \frac{5 π}{4}) = - \frac{\sqrt{2}}{2} \cdot (\cos (x) + \sin (x))$ is an identity