Trigonometry Examples

$4 \cos (45) - 9 \sin (45)$

Step 1

Simplify each term.

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Step 1.1

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

$4 \frac{\sqrt{2}}{2} - 9 \sin (45)$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{\sqrt{2}}{2} - 9 \sin (45)$

Step 1.2.2

Cancel the common factor.

$2 \cdot 2 \frac{\sqrt{2}}{2} - 9 \sin (45)$

Step 1.2.3

Rewrite the expression.

$2 \sqrt{2} - 9 \sin (45)$

Step 1.3

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

$2 \sqrt{2} - 9 \frac{\sqrt{2}}{2}$

Step 1.4

Combine $- 9$ and $\frac{\sqrt{2}}{2}$ .

$2 \sqrt{2} + \frac{- 9 \sqrt{2}}{2}$

Step 1.5

Move the negative in front of the fraction.

$2 \sqrt{2} - \frac{9 \sqrt{2}}{2}$

Step 2

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

$2 \sqrt{2} \cdot \frac{2}{2} - \frac{9 \sqrt{2}}{2}$

Step 3

Combine fractions.

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Step 3.1

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

$\frac{2 \sqrt{2} \cdot 2}{2} - \frac{9 \sqrt{2}}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{2 \sqrt{2} \cdot 2 - 9 \sqrt{2}}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $2$ .

$\frac{4 \sqrt{2} - 9 \sqrt{2}}{2}$

Step 4.2

Subtract $9 \sqrt{2}$ from $4 \sqrt{2}$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 3.53553390 \dots$