Trigonometry Examples

$\frac{\sin (315)}{1 + \cos (315)}$

Step 1

Simplify the numerator.

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Step 1.1

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.

$\frac{- \sin (45)}{1 + \cos (315)}$

Step 1.2

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

$\frac{- \frac{\sqrt{2}}{2}}{1 + \cos (315)}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Expand the denominator using the FOIL method.

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

Step 9

Simplify.

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

Step 10

Apply the distributive property.

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

Step 11

Move $2$ to the left of $\sqrt{2}$ .

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

Step 12

Multiply $\sqrt{2} (- \sqrt{2})$ .

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Add $1$ and $1$ .

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

Step 13

Simplify each term.

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Step 13.1

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

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Step 13.1.1

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

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

Step 13.1.2

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

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

Step 13.1.3

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

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

Step 13.1.4

Cancel the common factor of $2$ .

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Step 13.1.4.1

Cancel the common factor.

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

Step 13.1.4.2

Rewrite the expression.

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

Step 13.1.5

Evaluate the exponent.

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

Step 13.2

Multiply $- 1$ by $2$ .

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

Step 14

Simplify terms.

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Step 14.1

Cancel the common factor of $2 \sqrt{2} - 2$ and $2$ .

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Step 14.1.1

Factor $2$ out of $2 \sqrt{2}$ .

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

Step 14.1.2

Factor $2$ out of $- 2$ .

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

Step 14.1.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (- 1)$ .

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

Step 14.1.4

Cancel the common factors.

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Step 14.1.4.1

Factor $2$ out of $2$ .

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

Step 14.1.4.2

Cancel the common factor.

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

Step 14.1.4.3

Rewrite the expression.

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

Step 14.1.4.4

Divide $\sqrt{2} - 1$ by $1$ .

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

Step 14.2

Apply the distributive property.

$- \sqrt{2} - - 1$

Step 14.3

Multiply $- 1$ by $- 1$ .

$- \sqrt{2} + 1$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2} + 1$

Decimal Form:

$- 0.41421356 \dots$