Find the Exact Value sin(105)

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Trigonometry

$\sin (105)$

Split $105$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 + 60)$

Apply the sum of angles identity.

$\sin (45) \cos (60) + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{2} \cos (60) + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \frac{\sqrt{2}}{2} \sin (60)$

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

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

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Simplify each term.

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

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

$\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$

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

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

$\frac{\sqrt{2}}{4} + \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2}$

Combine using the product rule for radicals.

$\frac{\sqrt{2}}{4} + \frac{\sqrt{2 \cdot 3}}{2 \cdot 2}$

Multiply $2$ by $3$ .

$\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}$

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2} + \sqrt{6}}{4}$

Decimal Form:

$0.96592582 \dots$

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Name

Name	one billion six hundred twenty-seven million one hundred eighty-two thousand three hundred thirty-one

Interesting facts

1627182331 has 4 divisors, whose sum is 1650100464
The reverse of 1627182331 is 1332817261
Previous prime number is 71

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 34
Digital Root 7

Name

Name	one billion sixty-two million three hundred eighteen thousand nine hundred seven

Interesting facts

1062318907 has 8 divisors, whose sum is 1144288880
The reverse of 1062318907 is 7098132601
Previous prime number is 8971

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 37
Digital Root 1

Name

Name	one billion five hundred nineteen million three hundred ninety-five thousand four hundred eighteen

Interesting facts

1519395418 has 8 divisors, whose sum is 2280381012
The reverse of 1519395418 is 8145939151
Previous prime number is 1777

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 46
Digital Root 1

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