Trigonometry Examples

$\cos (60 + 45)$

Step 1

The complement of $\cos (60 + 45)$ is the angle that when added to $\cos (60 + 45)$ forms a right angle ( $90 °$ ).

$90 ° - \cos (60 + 45)$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Add $60$ and $45$ .

Step 2.2

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 2.2.1

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.

Step 2.2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

Step 2.2.3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

Step 2.2.4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

Step 2.2.5

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

Step 2.2.6

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

Step 2.2.7

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

Step 2.2.8

Simplify $- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

Tap for more steps...

Step 2.2.8.1

Simplify each term.

Tap for more steps...

Step 2.2.8.1.1

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

Tap for more steps...

Step 2.2.8.1.1.1

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

Step 2.2.8.1.1.2

Combine using the product rule for radicals.

Step 2.2.8.1.1.3

Multiply $3$ by $2$ .

Step 2.2.8.1.1.4

Multiply $2$ by $2$ .

Step 2.2.8.1.2

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

Tap for more steps...

Step 2.2.8.1.2.1

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

Step 2.2.8.1.2.2

Multiply $2$ by $2$ .

Step 2.2.8.2

Combine the numerators over the common denominator.

$90 + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 2.3

Multiply $- - \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 2.3.1

Multiply $- 1$ by $- 1$ .

Step 2.3.2

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $1$ .

$90 + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 3

To write $90$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$90 \cdot \frac{4}{4} + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 4

Combine $90$ and $\frac{4}{4}$ .

$\frac{90 \cdot 4}{4} + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Combine the numerators over the common denominator.

$\frac{90 \cdot 4 + \sqrt{6} - \sqrt{2}}{4 °}$

Step 5.2

Multiply $90$ by $4$ .

$\frac{360 + \sqrt{6} - \sqrt{2}}{4 °}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{360 + \sqrt{6} - \sqrt{2}}{4 °}$

Decimal Form:

$90.25881904 \dots$