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2*sin(a)=sin(3a)

Solution

2 \cdot sin (a) = sin (3 a)

Solution

a = 2 πn, a = π + 2 πn, a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn, a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 21 0^{\circ} + 36 0^{\circ} n, a = 33 0^{\circ} + 36 0^{\circ} n, a = 3 0^{\circ} + 36 0^{\circ} n, a = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (a) = sin (3 a)

Subtract

sin (3 a)

from both sides

2 sin (a) - sin (3 a) = 0

Rewrite using trig identities

- sin (3 a) + 2 sin (a)

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

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)

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

Simplify

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

- (3 sin (a) - 4 sin^{3} (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))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Add similar elements:

- 3 sin (a) + 2 sin (a) = - sin (a)

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

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

- sin (a) + 4 sin^{3} (a) = 0

Solve by substitution

- sin (a) + 4 sin^{3} (a) = 0

Let:

sin (a) = u

- u + 4 u^{3} = 0

- u + 4 u^{3} = 0 : u = 0, u = - \frac{1}{2}, u = \frac{1}{2}

- u + 4 u^{3} = 0

Factor

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

- u + 4 u^{3}

Factor out common term

u : u (4 u^{2} - 1)

4 u^{3} - u

Apply exponent rule:

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

u^{3} = u^{2} u

= 4 u^{2} u - u

Factor out common term

u

= u (4 u^{2} - 1)

= u (4 u^{2} - 1)

Factor

4 u^{2} - 1 : (2 u + 1) (2 u - 1)

4 u^{2} - 1

Rewrite

4 u^{2} - 1

(2 u)^{2} - 1^{2}

4 u^{2} - 1

Rewrite

4

2^{2}

= 2^{2} u^{2} - 1

Rewrite

1

1^{2}

= 2^{2} u^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - 1^{2}

= (2 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 u)^{2} - 1^{2} = (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u - 1)

= u (2 u + 1) (2 u - 1)

u (2 u + 1) (2 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 2 u + 1 = 0 or 2 u - 1 = 0

Solve

2 u + 1 = 0 : u = - \frac{1}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

\frac{2 u}{2} = \frac{- 1}{2}

Simplify

u = - \frac{1}{2}

u = - \frac{1}{2}

Solve

2 u - 1 = 0 : u = \frac{1}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

The solutions are

u = 0, u = - \frac{1}{2}, u = \frac{1}{2}

Substitute back

u = sin (a)

sin (a) = 0, sin (a) = - \frac{1}{2}, sin (a) = \frac{1}{2}

sin (a) = 0, sin (a) = - \frac{1}{2}, sin (a) = \frac{1}{2}

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

General solutions for

sin (a) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

sin (a) = - \frac{1}{2} : a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn

sin (a) = - \frac{1}{2}

General solutions for

sin (a) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn

a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn

sin (a) = \frac{1}{2} : a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn

sin (a) = \frac{1}{2}

General solutions for

sin (a) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn

a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn

Combine all the solutions

a = 2 πn, a = π + 2 πn, a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn, a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn

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