Expand the Trigonometric Expression sin(33 degrees +42 degrees )

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Trigonometry

$\sin (33 ° + 42 °)$

Add $33 °$ and $42 °$ .

$\sin (75)$

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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

$\sin (30 + 45)$

Apply the sum of angles identity.

$\sin (30) \cos (45) + \cos (30) \sin (45)$

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

$\frac{1}{2} \cos (45) + \cos (30) \sin (45)$

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

$\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)$

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

$\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)$

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

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

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

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Simplify each term.

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

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 forty million three hundred seventy-eight thousand six hundred five

Interesting facts

1640378605 has 16 divisors, whose sum is 1991127600
The reverse of 1640378605 is 5068730461
Previous prime number is 461

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 40
Digital Root 4

Name

Name	ninety-two million seven hundred twenty-one thousand three hundred fifteen

Interesting facts

92721315 has 16 divisors, whose sum is 100947840
The reverse of 92721315 is 51312729
Previous prime number is 73

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 30
Digital Root 3

Name

Name	five hundred nine million six hundred nineteen thousand two hundred seven

Interesting facts

509619207 has 4 divisors, whose sum is 679492280
The reverse of 509619207 is 702916905
Previous prime number is 3

Basic properties

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

Find the Exact Value cos(46)

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Find where Undefined/Discontinuous 2sin(3x)- square root of 3=0

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