Trigonometry Examples

$5 - \cos^{2} (60) - \cos^{2} (45)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

Step 1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.3

One to any power is one.

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

Step 1.4

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

$5 - \frac{1}{4} - \cos^{2} (45)$

Step 1.5

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

$5 - \frac{1}{4} - {(\frac{\sqrt{2}}{2})}^{2}$

Step 1.6

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

$5 - \frac{1}{4} - \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 1.7

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

Tap for more steps...

Step 1.7.1

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

$5 - \frac{1}{4} - \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 1.7.2

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

$5 - \frac{1}{4} - \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 1.7.3

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

$5 - \frac{1}{4} - \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 1.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.7.4.1

Cancel the common factor.

$5 - \frac{1}{4} - \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 1.7.4.2

Rewrite the expression.

$5 - \frac{1}{4} - \frac{2^{1}}{2^{2}}$

Step 1.7.5

Evaluate the exponent.

$5 - \frac{1}{4} - \frac{2}{2^{2}}$

Step 1.8

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

$5 - \frac{1}{4} - \frac{2}{4}$

Step 1.9

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

Tap for more steps...

Step 1.9.1

Factor $2$ out of $2$ .

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

Step 1.9.2

Cancel the common factors.

Tap for more steps...

Step 1.9.2.1

Factor $2$ out of $4$ .

$5 - \frac{1}{4} - \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.9.2.2

Cancel the common factor.

$5 - \frac{1}{4} - \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.9.2.3

Rewrite the expression.

$5 - \frac{1}{4} - \frac{1}{2}$

Step 2

Find the common denominator.

Tap for more steps...

Step 2.1

Write $5$ as a fraction with denominator $1$ .

$\frac{5}{1} - \frac{1}{4} - \frac{1}{2}$

Step 2.2

Multiply $\frac{5}{1}$ by $\frac{4}{4}$ .

$\frac{5}{1} \cdot \frac{4}{4} - \frac{1}{4} - \frac{1}{2}$

Step 2.3

Multiply $\frac{5}{1}$ by $\frac{4}{4}$ .

$\frac{5 \cdot 4}{4} - \frac{1}{4} - \frac{1}{2}$

Step 2.4

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

$\frac{5 \cdot 4}{4} - \frac{1}{4} - (\frac{1}{2} \cdot \frac{2}{2})$

Step 2.5

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

$\frac{5 \cdot 4}{4} - \frac{1}{4} - \frac{2}{2 \cdot 2}$

Step 2.6

Multiply $2$ by $2$ .

$\frac{5 \cdot 4}{4} - \frac{1}{4} - \frac{2}{4}$

Step 3

Combine the numerators over the common denominator.

$\frac{5 \cdot 4 - 1 - 2}{4}$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Multiply $5$ by $4$ .

$\frac{20 - 1 - 2}{4}$

Step 4.2

Subtract $1$ from $20$ .

$\frac{19 - 2}{4}$

Step 4.3

Subtract $2$ from $19$ .

$\frac{17}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{17}{4}$

Decimal Form:

$4.25$

Mixed Number Form:

$4 \frac{1}{4}$