Trigonometry Examples

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

Step 1

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

$4 \frac{\sqrt{2}}{2} \cos (45)$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $4$ .

$2 (2) \frac{\sqrt{2}}{2} \cos (45)$

Step 2.2

Cancel the common factor.

$2 \cdot 2 \frac{\sqrt{2}}{2} \cos (45)$

Step 2.3

Rewrite the expression.

$2 \sqrt{2} \cos (45)$

Step 3

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

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\sqrt{2} \sqrt{2}$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

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

Step 9.4.2

Rewrite the expression.

$2^{1}$

Step 9.5

Evaluate the exponent.

$2$