Trigonometry Examples

$\sqrt{\frac{1 - \cos (80)}{2}}$

Step 1

Rewrite $\sqrt{\frac{1 - \cos (80)}{2}}$ as $\frac{\sqrt{1 - \cos (80)}}{\sqrt{2}}$ .

$\frac{\sqrt{1 - \cos (80)}}{\sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{1 - \cos (80)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 - \cos (80)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{\sqrt{1 - \cos (80)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{2^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{\sqrt{1 - \cos (80)} \sqrt{2}}{2}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{(1 - \cos (80)) \cdot 2}}{2}$

Step 4.2

Evaluate $\cos (80)$ .

$\frac{\sqrt{(1 - 1 \cdot 0.17364817) \cdot 2}}{2}$

Step 4.3

Multiply $- 1$ by $0.17364817$ .

$\frac{\sqrt{(1 - 0.17364817) \cdot 2}}{2}$

Step 4.4

Subtract $0.17364817$ from $1$ .

$\frac{\sqrt{0.82635182 \cdot 2}}{2}$

Step 5

Multiply $0.82635182$ by $2$ .

$\frac{\sqrt{1.65270364}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt{\frac{1 - \cos (80)}{2}}$

Decimal Form:

$0.64278760 \dots$