Trigonometry Examples

$\frac{1 - \cos (45)}{\sin (45)}$

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.1.4

Cancel the common factor of $2$ .

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Step 10.1.4.1

Cancel the common factor.

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

Step 10.1.4.2

Rewrite the expression.

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

Step 10.1.5

Evaluate the exponent.

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

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

Cancel the common factor.

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

Step 10.2.2

Rewrite the expression.

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

Step 10.3

Multiply $- 1$ by $1$ .

$\sqrt{2} - 1$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2} - 1$

Decimal Form:

$0.41421356 \dots$