Verify the Identity sin(3x)=(sin(x))(4cos(x)^2-1)

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Trigonometry

$\sin (3 x) = (\sin (x)) (4 \cos^{2} (x) - 1)$

Start on the right side.

$(\sin (x)) (4 \cos^{2} (x) - 1)$

Apply Pythagorean identity in reverse.

$\sin (x) (4 (1 - \sin^{2} (x)) - 1)$

Apply the distributive property.

$\sin (x) (4 \cdot 1 + 4 (- \sin^{2} (x)) - 1)$

Simplify each term.

$\sin (x) (4 - 4 \sin^{2} (x) - 1)$

Apply the distributive property.

$\sin (x) \cdot 4 + \sin (x) (- 4 \sin^{2} (x)) + \sin (x) \cdot - 1$

Simplify.

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Simplify each term.

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Move $4$ to the left of $\sin (x)$ .

$4 \cdot \sin (x) + \sin (x) (- 4 {\sin (x)}^{2}) + \sin (x) \cdot - 1$

Multiply $\sin (x)$ by ${\sin (x)}^{2}$ by adding the exponents.

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Move ${\sin (x)}^{2}$ .

$4 \sin (x) + {\sin (x)}^{2} \sin (x) \cdot - 4 + \sin (x) \cdot - 1$

Multiply ${\sin (x)}^{2}$ by $\sin (x)$ .

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Raise $\sin (x)$ to the power of $1$ .

$4 \sin (x) + {\sin (x)}^{2} {\sin (x)}^{1} \cdot - 4 + \sin (x) \cdot - 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 \sin (x) + {\sin (x)}^{2 + 1} \cdot - 4 + \sin (x) \cdot - 1$

Add $2$ and $1$ .

$4 \sin (x) + {\sin (x)}^{3} \cdot - 4 + \sin (x) \cdot - 1$

Move $- 4$ to the left of ${\sin (x)}^{3}$ .

$4 \sin (x) - 4 \cdot {\sin (x)}^{3} + \sin (x) \cdot - 1$

Move $- 1$ to the left of $\sin (x)$ .

$4 \sin (x) - 4 {\sin (x)}^{3} - 1 \cdot \sin (x)$

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$4 \sin (x) - 4 {\sin (x)}^{3} - \sin (x)$

Subtract $\sin (x)$ from $4 \sin (x)$ .

$- 4 \sin^{3} (x) + 3 \sin (x)$

Apply the sine triple-angle identity.

$\sin (3 x)$

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (3 x) = (\sin (x)) (4 \cos^{2} (x) - 1)$ is an identity

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Name

Name	sixty-eight million four hundred thirty thousand one hundred ninety-seven

Interesting facts

68430197 has 4 divisors, whose sum is 74651136
The reverse of 68430197 is 79103486
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 38
Digital Root 2

Name

Name	three hundred eighty-two million forty-four thousand nine hundred twenty

Interesting facts

382044920 has 32 divisors, whose sum is 1547282088
The reverse of 382044920 is 029440283
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 32
Digital Root 5

Name

Name	one billion six hundred seventy-six million nine hundred forty-nine thousand seven hundred thirty-eight

Interesting facts

1676949738 has 16 divisors, whose sum is 3357984576
The reverse of 1676949738 is 8379496761
Previous prime number is 823

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 60
Digital Root 6

Graph v=11i+11j

Graph 3sin(7x)

Expand the Logarithmic Expression log base 6 of ((x^5)/y)^6

Convert to Interval Notation 11<=5x-4<=31

Convert to Trigonometric Form i^22