Trigonometry Examples

$- \sqrt{\frac{1 + \cos (225)}{2}}$

Step 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.

$- \sqrt{\frac{1 - \cos (45)}{2}}$

Step 2

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

$- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Write $1$ as a fraction with a common denominator.

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $2$ .

$- \sqrt{\frac{2 - \sqrt{2}}{4}}$

Step 6

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.38268343 \dots$