Trigonometry Examples

$h y π = \frac{1}{\cos (45)}$

Step 1

Simplify.

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Step 1.1

Convert from $\frac{1}{\cos (45)}$ to $\sec (45)$ .

$h y π = \sec (45)$

Step 1.2

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

$h y π = \frac{2}{\sqrt{2}}$

Step 1.3

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

$h y π = \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

$h y π = \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.2

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

$h y π = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.4.3

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

$h y π = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.4.4

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

$h y π = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.4.5

Add $1$ and $1$ .

$h y π = \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.4.6

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

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Step 1.4.6.1

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

$h y π = \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.4.6.2

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

$h y π = \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.4.6.3

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

$h y π = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$h y π = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.6.4.2

Rewrite the expression.

$h y π = \frac{2 \sqrt{2}}{2^{1}}$

Step 1.4.6.5

Evaluate the exponent.

$h y π = \frac{2 \sqrt{2}}{2}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$h y π = \frac{2 \sqrt{2}}{2}$

Step 1.5.2

Divide $\sqrt{2}$ by $1$ .

$h y π = \sqrt{2}$

Step 2

Divide each term in $h y π = \sqrt{2}$ by $y π$ and simplify.

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Step 2.1

Divide each term in $h y π = \sqrt{2}$ by $y π$ .

$\frac{h y π}{y π} = \frac{\sqrt{2}}{y π}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $y$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{h y π}{y π} = \frac{\sqrt{2}}{y π}$

Step 2.2.1.2

Rewrite the expression.

$\frac{h π}{π} = \frac{\sqrt{2}}{y π}$

Step 2.2.2

Cancel the common factor of $π$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{h π}{π} = \frac{\sqrt{2}}{y π}$

Step 2.2.2.2

Divide $h$ by $1$ .

$h = \frac{\sqrt{2}}{y π}$