Algebra Examples

$\sin (45 °) + \sin (135 °) + \sin (225 °) + \sin (315 °)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (45 °)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + \sin (135 °) + \sin (225 °) + \sin (315 °)$

Step 1.2

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

$\frac{\sqrt{2}}{2} + \sin (45) + \sin (225 °) + \sin (315 °)$

Step 1.3

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \sin (225 °) + \sin (315 °)$

Step 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 third quadrant.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \sin (45) + \sin (315 °)$

Step 1.5

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + \sin (315 °)$

Step 1.6

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.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \sin (45)$

Step 1.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{2} - \sqrt{2} - \sqrt{2}}{2}$

Step 2.2

Add $\sqrt{2}$ and $\sqrt{2}$ .

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

Step 2.3

Subtract $\sqrt{2}$ from $2 \sqrt{2}$ .

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

Step 2.4

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{2}$

Step 2.5

Divide $0$ by $2$ .

$0$