Trigonometry Examples

$\cos (\frac{7 π}{4} + x) + \sin (\frac{5 π}{4} + x) = 0$

Step 1

Start on the left side.

$\cos (\frac{7 π}{4} + x) + \sin (\frac{5 π}{4} + x)$

Step 2

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

$\cos (\frac{7 π}{4}) \cos (x) - \sin (\frac{7 π}{4}) \sin (x) + \sin (\frac{5 π}{4} + x)$

Step 3

Apply the sum of angles identity.

$\cos (\frac{7 π}{4}) \cos (x) - \sin (\frac{7 π}{4}) \sin (x) + \sin (\frac{5 π}{4}) \cos (x) + \cos (\frac{5 π}{4}) \sin (x)$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\cos (\frac{π}{4}) \cos (x) - \sin (\frac{7 π}{4}) \sin (x) + \sin (\frac{5 π}{4}) \cos (x) + \cos (\frac{5 π}{4}) \sin (x)$

Step 4.1.2

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

$\frac{\sqrt{2}}{2} \cos (x) - \sin (\frac{7 π}{4}) \sin (x) + \sin (\frac{5 π}{4}) \cos (x) + \cos (\frac{5 π}{4}) \sin (x)$

Step 4.1.3

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

$\frac{\sqrt{2} \cos (x)}{2} - \sin (\frac{7 π}{4}) \sin (x) + \sin (\frac{5 π}{4}) \cos (x) + \cos (\frac{5 π}{4}) \sin (x)$

Step 4.1.4

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 fourth quadrant.

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

Step 4.1.5

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

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

Step 4.1.6

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 4.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{2} \cos (x)}{2} + 1 \frac{\sqrt{2}}{2} \sin (x) + \sin (\frac{5 π}{4}) \cos (x) + \cos (\frac{5 π}{4}) \sin (x)$

Step 4.1.6.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

Step 4.1.7

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

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

Step 4.1.8

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.

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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.

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

Step 4.1.12

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

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

Step 4.1.13

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

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Combine the opposite terms in $\sqrt{2} \cos (x) + \sqrt{2} \sin (x) - \cos (x) \sqrt{2} - \sin (x) \sqrt{2}$ .

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Step 4.3.1

Reorder the factors in the terms $\sqrt{2} \cos (x)$ and $- \cos (x) \sqrt{2}$ .

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

Step 4.3.2

Subtract $\sqrt{2} \cos (x)$ from $\sqrt{2} \cos (x)$ .

$\frac{0 + \sqrt{2} \sin (x) - \sin (x) \sqrt{2}}{2}$

Step 4.3.3

Add $0$ and $\sqrt{2} \sin (x)$ .

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

Step 4.3.4

Reorder the factors in the terms $\sqrt{2} \sin (x)$ and $- \sin (x) \sqrt{2}$ .

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

Step 4.3.5

Subtract $\sqrt{2} \sin (x)$ from $\sqrt{2} \sin (x)$ .

$\frac{0}{2}$

Step 4.4

Divide $0$ by $2$ .

$0$

Step 5

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

$\cos (\frac{7 π}{4} + x) + \sin (\frac{5 π}{4} + x) = 0$ is an identity