Finite Math Examples

$\sin (105) (x b a^{5 r})$

Step 1

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

Tap for more steps...

Step 1.1

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

$\sin (75) (x b a^{5 r})$

Step 1.2

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

$\sin (30 + 45) (x b a^{5 r})$

Step 1.3

Apply the sum of angles identity.

$(\sin (30) \cos (45) + \cos (30) \sin (45)) (x b a^{5 r})$

Step 1.4

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

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) (x b a^{5 r})$

Step 1.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) (x b a^{5 r})$

Step 1.6

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) (x b a^{5 r})$

Step 1.7

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) (x b a^{5 r})$

Step 1.8

Simplify $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 1.8.1

Simplify each term.

Tap for more steps...

Step 1.8.1.1

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

Tap for more steps...

Step 1.8.1.1.1

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) (x b a^{5 r})$

Step 1.8.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) (x b a^{5 r})$

Step 1.8.1.2

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

Tap for more steps...

Step 1.8.1.2.1

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) (x b a^{5 r})$

Step 1.8.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) (x b a^{5 r})$

Step 1.8.1.2.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) (x b a^{5 r})$

Step 1.8.1.2.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) (x b a^{5 r})$

Step 1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} (x b a^{5 r})$

Step 2

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} (x b a^{5 r})$ .

Tap for more steps...

Step 2.1

Combine $x$ and $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

$\frac{x (\sqrt{2} + \sqrt{6})}{4} (b a^{5 r})$

Step 2.2

Combine $b$ and $\frac{x (\sqrt{2} + \sqrt{6})}{4}$ .

$\frac{b (x (\sqrt{2} + \sqrt{6}))}{4} a^{5 r}$

Step 2.3

Combine $\frac{b (x (\sqrt{2} + \sqrt{6}))}{4}$ and $a^{5 r}$ .

$\frac{b (x (\sqrt{2} + \sqrt{6})) a^{5 r}}{4}$

$\frac{b x (\sqrt{2} + \sqrt{6}) a^{5 r}}{4}$