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	four hundred thirty-seven million seven hundred thirteen thousand nine hundred twenty-three

Interesting facts

437713923 has 4 divisors, whose sum is 583618568
The reverse of 437713923 is 329317734
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 39
Digital Root 3

Name

Name	one billion eight hundred eighty-four million seven hundred forty thousand two hundred twenty-five

Interesting facts

1884740225 has 32 divisors, whose sum is 2803196160
The reverse of 1884740225 is 5220474881
Previous prime number is 37

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	two billion fifty million two hundred ninety-eight thousand seven hundred thirty-one

Interesting facts

2050298731 has 4 divisors, whose sum is 2050996464
The reverse of 2050298731 is 1378920502
Previous prime number is 2951

Basic properties

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

Graph y=x^2-12x+2

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Graph y=-3+4cos(pi(x-2))

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