Trigonometry Examples

$\cos (112.5) \cos (67.5)$

Step 1

The exact value of $\cos (112.5)$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

Step 1.1

Rewrite $112.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{225}{2}) \cos (67.5)$

Step 1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$(\pm \sqrt{\frac{1 + \cos (225)}{2}}) \cos (67.5)$

Step 1.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$- \sqrt{\frac{1 + \cos (225)}{2}} \cos (67.5)$

Step 1.4

Simplify $- \sqrt{\frac{1 + \cos (225)}{2}}$ .

Tap for more steps...

Step 1.4.1

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.

$- \sqrt{\frac{1 - \cos (45)}{2}} \cos (67.5)$

Step 1.4.2

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

$- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \cos (67.5)$

Step 1.4.3

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

$- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} \cos (67.5)$

Step 1.4.4

Combine the numerators over the common denominator.

$- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} \cos (67.5)$

Step 1.4.5

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} \cos (67.5)$

Step 1.4.6

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

Tap for more steps...

Step 1.4.6.1

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

$- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} \cos (67.5)$

Step 1.4.6.2

Multiply $2$ by $2$ .

$- \sqrt{\frac{2 - \sqrt{2}}{4}} \cos (67.5)$

Step 1.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} \cos (67.5)$

Step 1.4.8

Simplify the denominator.

Tap for more steps...

Step 1.4.8.1

Rewrite $4$ as $2^{2}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} \cos (67.5)$

Step 1.4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cos (67.5)$

Step 2

The exact value of $\cos (67.5)$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

Step 2.1

Rewrite $67.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cos (\frac{135}{2})$

Step 2.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} (\pm \sqrt{\frac{1 + \cos (135)}{2}})$

Step 2.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{1 + \cos (135)}{2}}$

Step 2.4

Simplify $\sqrt{\frac{1 + \cos (135)}{2}}$ .

Tap for more steps...

Step 2.4.1

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 second quadrant.

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

Step 2.4.2

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$

Step 2.4.3

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

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

Step 2.4.4

Combine the numerators over the common denominator.

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

Step 2.4.5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}$

Step 2.4.6

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

Tap for more steps...

Step 2.4.6.1

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

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}$

Step 2.4.6.2

Multiply $2$ by $2$ .

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

Step 2.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$

Step 2.4.8

Simplify the denominator.

Tap for more steps...

Step 2.4.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \frac{\sqrt{2 - \sqrt{2}}}{2}$

Step 3

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

Tap for more steps...

Step 3.1

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

$- \frac{\sqrt{2 - \sqrt{2}} \sqrt{2 - \sqrt{2}}}{2 \cdot 2}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$- \frac{{\sqrt{2 - \sqrt{2}}}^{1 + 1}}{2 \cdot 2}$

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Multiply $2$ by $2$ .

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Simplify.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.14644660 \dots$