Precalculus Examples

$\sin (45) \cos (45) \tan (45)$

Step 1

To convert degrees to radians, multiply by

$\frac{π}{180 °}$ , since a full circle is

$360 °$ or

$2 π$ radians.

Step 2

The exact value of

$\sin (45)$ is

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

$\frac{\sqrt{2}}{2} \cdot (\cos (45) \tan (45)) \cdot \frac{π}{180}$ radians

Step 3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.1

Add parentheses.

$\frac{\sqrt{2}}{2} \cdot (\cos (45) \tan (45)) \cdot \frac{π}{180}$ radians

Step 3.2

Reorder

$\cos (45)$ and

$\tan (45)$ .

$\frac{\sqrt{2}}{2} \cdot (\tan (45) \cos (45)) \cdot \frac{π}{180}$ radians

Step 3.3

Rewrite

$\frac{\sqrt{2}}{2} \cos (45) \tan (45) \cdot \frac{π}{180}$ in terms of sines and cosines.

$\frac{\sqrt{2}}{2} \cdot (\frac{\sin (45)}{\cos (45)} \cdot \cos (45)) \cdot \frac{π}{180}$ radians

Step 3.4

Cancel the common factors.

$\frac{\sqrt{2}}{2} \cdot \sin (45) \cdot \frac{π}{180}$ radians

Step 4

The exact value of

$\sin (45)$ is

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} \cdot \frac{π}{180}$ radians

Step 5

Multiply

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

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Step 5.1

Multiply

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

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

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} \cdot \frac{π}{180}$ radians

Step 5.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} \cdot \frac{π}{180}$ radians

Step 5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} \cdot \frac{π}{180}$ radians

Step 5.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} \cdot \frac{π}{180}$ radians

Step 5.5

Add

$1$ and

$1$ .

$\frac{{\sqrt{2}}^{2}}{2 \cdot 2} \cdot \frac{π}{180}$ radians

Step 5.6

Multiply

$2$ by

$2$ .

$\frac{{\sqrt{2}}^{2}}{4} \cdot \frac{π}{180}$ radians

Step 6

Combine.

$\frac{{\sqrt{2}}^{2} π}{4 \cdot 180}$ radians

Step 7

Rewrite

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

$2$ .

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Step 7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

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

$\frac{{(2^{\frac{1}{2}})}^{2} π}{4 \cdot 180}$ radians

Step 7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{\frac{1}{2} \cdot 2} π}{4 \cdot 180}$ radians

Step 7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{2^{\frac{2}{2}} π}{4 \cdot 180}$ radians

Step 7.4

Cancel the common factor of

$2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}} π}{4 \cdot 180}$ radians

Step 7.4.2

Rewrite the expression.

$\frac{2 π}{4 \cdot 180}$ radians

Step 7.5

Evaluate the exponent.

$\frac{2 π}{4 \cdot 180}$ radians

Step 8

Multiply

$4$ by

$180$ .

$\frac{2 π}{720}$ radians

Step 9

Cancel the common factor of

$2$ and

$720$ .

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Step 9.1

Factor

$2$ out of

$2 π$ .

$\frac{2 (π)}{720}$ radians

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor

$2$ out of

$720$ .

$\frac{2 π}{2 \cdot 360}$ radians

Step 9.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 360}$ radians

Step 9.2.3

Rewrite the expression.

$\frac{π}{360}$ radians