Trigonometry Examples

$\sqrt{\frac{1 + \cos (75)}{2}}$

Step 1

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\sqrt{\frac{1 + \cos (30 + 45)}{2}}$

Step 1.2

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

$\sqrt{\frac{1 + \cos (30) \cos (45) - \sin (30) \sin (45)}{2}}$

Step 1.3

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)}{2}}$

Step 1.4

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

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)}{2}}$

Step 1.5

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)}{2}}$

Step 1.6

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

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{2}}$

Step 1.7

Simplify $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.7.1.1.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{1 + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{2}}$

Step 1.7.1.1.2

Combine using the product rule for radicals.

$\sqrt{\frac{1 + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{2}}$

Step 1.7.1.1.3

Multiply $3$ by $2$ .

$\sqrt{\frac{1 + \frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{2}}$

Step 1.7.1.1.4

Multiply $2$ by $2$ .

$\sqrt{\frac{1 + \frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{2}}$

Step 1.7.1.2

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

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Step 1.7.1.2.1

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

$\sqrt{\frac{1 + \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{2}}$

Step 1.7.1.2.2

Multiply $2$ by $2$ .

$\sqrt{\frac{1 + \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{2}}$

Step 1.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + \frac{\sqrt{6} - \sqrt{2}}{4}}{2}}$

Step 2

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

$\sqrt{\frac{\frac{4}{4} + \frac{\sqrt{6} - \sqrt{2}}{4}}{2}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{4 + \sqrt{6} - \sqrt{2}}{4}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{4 + \sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{4 + \sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{4 + \sqrt{6} - \sqrt{2}}{4}$ by $\frac{1}{2}$ .

$\sqrt{\frac{4 + \sqrt{6} - \sqrt{2}}{4 \cdot 2}}$

Step 5.2

Multiply $4$ by $2$ .

$\sqrt{\frac{4 + \sqrt{6} - \sqrt{2}}{8}}$

Step 6

Rewrite $\sqrt{\frac{4 + \sqrt{6} - \sqrt{2}}{8}}$ as $\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{\sqrt{8}}$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{\sqrt{8}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{\sqrt{4 (2)}}$

Step 7.1.2

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{\sqrt{2^{2} \cdot 2}}$

Step 7.2

Pull terms out from under the radical.

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{2 \sqrt{2}}$

Step 8

Multiply $\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 9.2

Move $\sqrt{2}$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 9.3

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 9.4

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 9.5

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 9.6

Add $1$ and $1$ .

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 9.7

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

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Step 9.7.1

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 9.7.2

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 9.7.3

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

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 9.7.4.2

Rewrite the expression.

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \cdot 2^{1}}$

Step 9.7.5

Evaluate the exponent.

$\frac{\sqrt{4 + \sqrt{6} - \sqrt{2}} \sqrt{2}}{2 \cdot 2}$

Step 10

Combine using the product rule for radicals.

$\frac{\sqrt{(4 + \sqrt{6} - \sqrt{2}) \cdot 2}}{2 \cdot 2}$

Step 11

Multiply $2$ by $2$ .

$\frac{\sqrt{(4 + \sqrt{6} - \sqrt{2}) \cdot 2}}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(4 + \sqrt{6} - \sqrt{2}) \cdot 2}}{4}$

Decimal Form:

$0.79335334 \dots$