Trigonometry Examples

$\sqrt{6} \cos (315) + i \sin (315)$

Step 1

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

$\sqrt{6} \cos (45) + i \sin (315)$

Step 2

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

$\sqrt{6} \frac{\sqrt{2}}{2} + i \sin (315)$

Step 3

Multiply $\sqrt{6} \frac{\sqrt{2}}{2}$ .

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Step 3.1

Combine $\sqrt{6}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} \sqrt{2}}{2} + i \sin (315)$

Step 3.2

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2}}{2} + i \sin (315)$

Step 3.3

Multiply $6$ by $2$ .

$\frac{\sqrt{12}}{2} + i \sin (315)$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)}}{2} + i \sin (315)$

Step 4.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3}}{2} + i \sin (315)$

Step 4.2

Pull terms out from under the radical.

$\frac{2 \sqrt{3}}{2} + i \sin (315)$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{2} + i \sin (315)$

Step 5.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3} + i \sin (315)$

Step 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.

$\sqrt{3} + i (- \sin (45))$

Step 7

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

$\sqrt{3} + i (- \frac{\sqrt{2}}{2})$

Step 8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$\sqrt{3} - \frac{i \sqrt{2}}{2}$