Calculus Examples

$\cot^{2} (135) - \sin (210) + 5 \cos^{2} (225)$

Step 1

Simplify each term.

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Step 1.1

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

${(- \cot (45))}^{2} - \sin (210) + 5 \cos^{2} (225)$

Step 1.2

The exact value of $\cot (45)$ is $1$ .

${(- 1 \cdot 1)}^{2} - \sin (210) + 5 \cos^{2} (225)$

Step 1.3

Multiply $- 1$ by $1$ .

${(- 1)}^{2} - \sin (210) + 5 \cos^{2} (225)$

Step 1.4

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

$1 - \sin (210) + 5 \cos^{2} (225)$

Step 1.5

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.

$1 - - \sin (30) + 5 \cos^{2} (225)$

Step 1.6

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$1 - - \frac{1}{2} + 5 \cos^{2} (225)$

Step 1.7

Multiply $- - \frac{1}{2}$ .

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Step 1.7.1

Multiply $- 1$ by $- 1$ .

$1 + 1 (\frac{1}{2}) + 5 \cos^{2} (225)$

Step 1.7.2

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

$1 + \frac{1}{2} + 5 \cos^{2} (225)$

Step 1.8

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.

$1 + \frac{1}{2} + 5 {(- \cos (45))}^{2}$

Step 1.9

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

$1 + \frac{1}{2} + 5 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.10.1

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

$1 + \frac{1}{2} + 5 ({(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.10.2

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

$1 + \frac{1}{2} + 5 ({(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 1.11

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

$1 + \frac{1}{2} + 5 (1 \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 1.12

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

$1 + \frac{1}{2} + 5 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 1.13

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

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Step 1.13.1

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

$1 + \frac{1}{2} + 5 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 1.13.2

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

$1 + \frac{1}{2} + 5 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 1.13.3

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

$1 + \frac{1}{2} + 5 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 1.13.4

Cancel the common factor of $2$ .

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Step 1.13.4.1

Cancel the common factor.

$1 + \frac{1}{2} + 5 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 1.13.4.2

Rewrite the expression.

$1 + \frac{1}{2} + 5 \frac{2^{1}}{2^{2}}$

Step 1.13.5

Evaluate the exponent.

$1 + \frac{1}{2} + 5 \frac{2}{2^{2}}$

Step 1.14

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

$1 + \frac{1}{2} + 5 (\frac{2}{4})$

Step 1.15

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

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Step 1.15.1

Factor $2$ out of $2$ .

$1 + \frac{1}{2} + 5 \frac{2 (1)}{4}$

Step 1.15.2

Cancel the common factors.

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Step 1.15.2.1

Factor $2$ out of $4$ .

$1 + \frac{1}{2} + 5 \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.15.2.2

Cancel the common factor.

$1 + \frac{1}{2} + 5 \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.15.2.3

Rewrite the expression.

$1 + \frac{1}{2} + 5 (\frac{1}{2})$

Step 1.16

Combine $5$ and $\frac{1}{2}$ .

$1 + \frac{1}{2} + \frac{5}{2}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$1 + \frac{1 + 5}{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $5$ .

$1 + \frac{6}{2}$

Step 2.2.2

Divide $6$ by $2$ .

$1 + 3$

Step 2.2.3

Add $1$ and $3$ .

$4$