Is sin(3A)=3sin(A)-4sin^3(A) ?

The answer to whether sin(3A)=3sin(A)-4sin^3(A) is True

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Popular Trigonometry >

prove sin(3A)=3sin(A)-4sin^3(A)

Solution

prove

sin (3 A) = 3 sin (A) - 4 sin^{3} (A)

Solution

True

Solution steps

sin (3 A) = 3 sin (A) - 4 sin^{3} (A)

Manipulating left side

sin (3 A)

Rewrite using trig identities

sin (3 A)

Rewrite as

= sin (2 A + A)

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (2 A) cos (A) + cos (2 A) sin (A)

Use the Double Angle identity:

sin (2 A) = 2 sin (A) cos (A)

= cos (2 A) sin (A) + cos (A) 2 sin (A) cos (A)

Simplify

cos (2 A) sin (A) + cos (A) \cdot 2 sin (A) cos (A) : sin (A) cos (2 A) + 2 cos^{2} (A) sin (A)

cos (2 A) sin (A) + cos (A) 2 sin (A) cos (A)

cos (A) \cdot 2 sin (A) cos (A) = 2 cos^{2} (A) sin (A)

cos (A) 2 sin (A) cos (A)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (A) cos (A) = cos^{1 + 1} (A)

= 2 sin (A) cos^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= 2 sin (A) cos^{2} (A)

= sin (A) cos (2 A) + 2 cos^{2} (A) sin (A)

= sin (A) cos (2 A) + 2 cos^{2} (A) sin (A)

= sin (A) cos (2 A) + 2 cos^{2} (A) sin (A)

Use the Double Angle identity:

cos (2 A) = 1 - 2 sin^{2} (A)

= (1 - 2 sin^{2} (A)) sin (A) + 2 cos^{2} (A) sin (A)

Use the Pythagorean identity:

cos^{2} (A) + sin^{2} (A) = 1

cos^{2} (A) = 1 - sin^{2} (A)

= (1 - 2 sin^{2} (A)) sin (A) + 2 (1 - sin^{2} (A)) sin (A)

Expand

(1 - 2 sin^{2} (A)) sin (A) + 2 (1 - sin^{2} (A)) sin (A) : - 4 sin^{3} (A) + 3 sin (A)

(1 - 2 sin^{2} (A)) sin (A) + 2 (1 - sin^{2} (A)) sin (A)

= sin (A) (1 - 2 sin^{2} (A)) + 2 sin (A) (1 - sin^{2} (A))

Expand

sin (A) (1 - 2 sin^{2} (A)) : sin (A) - 2 sin^{3} (A)

sin (A) (1 - 2 sin^{2} (A))

Apply the distributive law:

a (b - c) = ab - a c

a = sin (A), b = 1, c = 2 sin^{2} (A)

= sin (A) 1 - sin (A) 2 sin^{2} (A)

= 1 sin (A) - 2 sin^{2} (A) sin (A)

Simplify

1 \cdot sin (A) - 2 sin^{2} (A) sin (A) : sin (A) - 2 sin^{3} (A)

1 sin (A) - 2 sin^{2} (A) sin (A)

1 \cdot sin (A) = sin (A)

1 sin (A)

Multiply:

1 \cdot sin (A) = sin (A)

= sin (A)

2 sin^{2} (A) sin (A) = 2 sin^{3} (A)

2 sin^{2} (A) sin (A)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (A) sin (A) = sin^{2 + 1} (A)

= 2 sin^{2 + 1} (A)

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (A)

= sin (A) - 2 sin^{3} (A)

= sin (A) - 2 sin^{3} (A)

= sin (A) - 2 sin^{3} (A) + 2 (1 - sin^{2} (A)) sin (A)

Expand

2 sin (A) (1 - sin^{2} (A)) : 2 sin (A) - 2 sin^{3} (A)

2 sin (A) (1 - sin^{2} (A))

Apply the distributive law:

a (b - c) = ab - a c

a = 2 sin (A), b = 1, c = sin^{2} (A)

= 2 sin (A) 1 - 2 sin (A) sin^{2} (A)

= 2 \cdot 1 sin (A) - 2 sin^{2} (A) sin (A)

Simplify

2 \cdot 1 \cdot sin (A) - 2 sin^{2} (A) sin (A) : 2 sin (A) - 2 sin^{3} (A)

2 \cdot 1 sin (A) - 2 sin^{2} (A) sin (A)

2 \cdot 1 \cdot sin (A) = 2 sin (A)

2 \cdot 1 sin (A)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (A)

2 sin^{2} (A) sin (A) = 2 sin^{3} (A)

2 sin^{2} (A) sin (A)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (A) sin (A) = sin^{2 + 1} (A)

= 2 sin^{2 + 1} (A)

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (A)

= 2 sin (A) - 2 sin^{3} (A)

= 2 sin (A) - 2 sin^{3} (A)

= sin (A) - 2 sin^{3} (A) + 2 sin (A) - 2 sin^{3} (A)

Simplify

sin (A) - 2 sin^{3} (A) + 2 sin (A) - 2 sin^{3} (A) : - 4 sin^{3} (A) + 3 sin (A)

sin (A) - 2 sin^{3} (A) + 2 sin (A) - 2 sin^{3} (A)

Group like terms

= - 2 sin^{3} (A) - 2 sin^{3} (A) + sin (A) + 2 sin (A)

Add similar elements:

- 2 sin^{3} (A) - 2 sin^{3} (A) = - 4 sin^{3} (A)

= - 4 sin^{3} (A) + sin (A) + 2 sin (A)

Add similar elements:

sin (A) + 2 sin (A) = 3 sin (A)

= - 4 sin^{3} (A) + 3 sin (A)

= - 4 sin^{3} (A) + 3 sin (A)

= - 4 sin^{3} (A) + 3 sin (A)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is sin(3A)=3sin(A)-4sin^3(A) ?
The answer to whether sin(3A)=3sin(A)-4sin^3(A) is True