Calculus Examples

$\sec^{2} (\frac{3 π}{4})$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.

${(- \sec (\frac{π}{4}))}^{2}$

Step 2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

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

${(- \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}$

Add $1$ and $1$ .

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

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

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

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

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

${(- \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}$

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 5

Cancel the common factor of $2$ .

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Cancel the common factor.

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

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

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

Step 6

Simplify the expression.

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Apply the product rule to $- \sqrt{2}$ .

${(- 1)}^{2} {\sqrt{2}}^{2}$

Raise $- 1$ to the power of $2$ .

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

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

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

Step 7

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$2^{1}$

Evaluate the exponent.

$2$