Algebra Examples

$\frac{\sin (\frac{3 π}{2}) - 5 \cos (\frac{13 π}{4})}{\tan (\frac{25 π}{4})}$

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.

$\frac{- \sin (\frac{π}{2}) - 5 \cos (\frac{13 π}{4})}{\tan (\frac{25 π}{4})}$

Step 1.2

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

$\frac{- 1 \cdot 1 - 5 \cos (\frac{13 π}{4})}{\tan (\frac{25 π}{4})}$

Step 1.3

Multiply $- 1$ by $1$ .

$\frac{- 1 - 5 \cos (\frac{13 π}{4})}{\tan (\frac{25 π}{4})}$

Step 1.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{- 1 - 5 \cos (\frac{5 π}{4})}{\tan (\frac{25 π}{4})}$

Step 1.5

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.

$\frac{- 1 - 5 (- \cos (\frac{π}{4}))}{\tan (\frac{25 π}{4})}$

Step 1.6

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

$\frac{- 1 - 5 (- \frac{\sqrt{2}}{2})}{\tan (\frac{25 π}{4})}$

Step 1.7

Multiply $- 5 (- \frac{\sqrt{2}}{2})$ .

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Step 1.7.1

Multiply $- 1$ by $- 5$ .

$\frac{- 1 + 5 \frac{\sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 1.7.2

Combine $5$ and $\frac{\sqrt{2}}{2}$ .

$\frac{- 1 + \frac{5 \sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 1.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 1 \cdot \frac{2}{2} + \frac{5 \sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 1.9

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

$\frac{\frac{- 1 \cdot 2}{2} + \frac{5 \sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 1.10

Combine the numerators over the common denominator.

$\frac{\frac{- 1 \cdot 2 + 5 \sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 1.11

Multiply $- 1$ by $2$ .

$\frac{\frac{- 2 + 5 \sqrt{2}}{2}}{\tan (\frac{25 π}{4})}$

Step 2

Simplify the denominator.

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Step 2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\frac{- 2 + 5 \sqrt{2}}{2}}{\tan (\frac{π}{4})}$

Step 2.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\frac{- 2 + 5 \sqrt{2}}{2}}{1}$

Step 3

Simplify with factoring out.

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Step 3.1

Divide $\frac{- 2 + 5 \sqrt{2}}{2}$ by $1$ .

$\frac{- 2 + 5 \sqrt{2}}{2}$

Step 3.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) + 5 \sqrt{2}}{2}$

Step 3.3

Factor $- 1$ out of $5 \sqrt{2}$ .

$\frac{- 1 (2) - (- 5 \sqrt{2})}{2}$

Step 3.4

Factor $- 1$ out of $- 1 (2) - (- 5 \sqrt{2})$ .

$\frac{- 1 (2 - 5 \sqrt{2})}{2}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{2 - 5 \sqrt{2}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 - 5 \sqrt{2}}{2}$

Decimal Form:

$2.53553390 \dots$