Calculus Examples

$(- \cos (\frac{π}{2} + \frac{2 π}{3})) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 1

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \cos (\frac{π}{2} \cdot \frac{3}{3} + \frac{2 π}{3}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \cos (\frac{π}{2} \cdot \frac{3}{3} + \frac{2 π}{3} \cdot \frac{2}{2}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$- \cos (\frac{π \cdot 3}{2 \cdot 3} + \frac{2 π}{3} \cdot \frac{2}{2}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 3.2

Multiply $2$ by $3$ .

$- \cos (\frac{π \cdot 3}{6} + \frac{2 π}{3} \cdot \frac{2}{2}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 3.3

Multiply $\frac{2 π}{3}$ by $\frac{2}{2}$ .

$- \cos (\frac{π \cdot 3}{6} + \frac{2 π \cdot 2}{3 \cdot 2}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 3.4

Multiply $3$ by $2$ .

$- \cos (\frac{π \cdot 3}{6} + \frac{2 π \cdot 2}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 4

Combine the numerators over the common denominator.

$- \cos (\frac{π \cdot 3 + 2 π \cdot 2}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 5

Simplify the numerator.

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Step 5.1

Move $3$ to the left of $π$ .

$- \cos (\frac{3 \cdot π + 2 π \cdot 2}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 5.2

Multiply $2$ by $2$ .

$- \cos (\frac{3 π + 4 π}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 5.3

Add $3 π$ and $4 π$ .

$- \cos (\frac{7 π}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 6

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.

$- - \cos (\frac{π}{6}) - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 7

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- - \frac{\sqrt{3}}{2} - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 8

Multiply $- - \frac{\sqrt{3}}{2}$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3}}{2} - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 8.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{\sqrt{3}}{2} - (\cos (- \frac{π}{4}) - \frac{π}{2})$

Step 9

Simplify each term.

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Step 9.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sqrt{3}}{2} - (\cos (\frac{7 π}{4}) - \frac{π}{2})$

Step 9.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sqrt{3}}{2} - (\cos (\frac{π}{4}) - \frac{π}{2})$

Step 9.3

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

$\frac{\sqrt{3}}{2} - (\frac{\sqrt{2}}{2} - \frac{π}{2})$

Step 10

Apply the distributive property.

$\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} - - \frac{π}{2}$

Step 11

Multiply $- - \frac{π}{2}$ .

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Step 11.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} + 1 \frac{π}{2}$

Step 11.2

Multiply $\frac{π}{2}$ by $1$ .

$\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} + \frac{π}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} + \frac{π}{2}$

Decimal Form:

$1.72971494 \dots$