Precalculus Examples

$\sin (\frac{13 π}{4}) \cos (\frac{13 π}{4})$

Step 1

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

$\sin (\frac{5 π}{4}) \cos (\frac{13 π}{4})$

Step 2

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 third quadrant.

$- \sin (\frac{π}{4}) \cos (\frac{13 π}{4})$

Step 3

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

$- \frac{\sqrt{2}}{2} \cos (\frac{13 π}{4})$

Step 4

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

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

Step 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.

$- \frac{\sqrt{2}}{2} (- \cos (\frac{π}{4}))$

Step 6

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

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

Step 7

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

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Step 7.1

Multiply $- 1$ by $- 1$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

Add $1$ and $1$ .

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

Step 7.8

Multiply $2$ by $2$ .

$\frac{{\sqrt{2}}^{2}}{4}$

Step 8

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

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Step 8.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{4}$

Step 8.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{4}$

Step 8.3

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

$\frac{2^{\frac{2}{2}}}{4}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{4}$

Step 8.4.2

Rewrite the expression.

$\frac{2^{1}}{4}$

Step 8.5

Evaluate the exponent.

$\frac{2}{4}$

Step 9

Cancel the common factor of $2$ and $4$ .

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Step 9.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$