What is the general solution for sin^2(a)=2sin(a)cos(a) ?

The general solution for sin^2(a)=2sin(a)cos(a) is a=2pin,a=pi+2pin,a=1.10714…+pin

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sin^2(a)=2sin(a)cos(a)

Solution

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

Solution

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

2 sin (a) cos (a)

from both sides

sin^{2} (a) - 2 sin (a) cos (a) = 0

Factor

sin^{2} (a) - 2 sin (a) cos (a) : sin (a) (sin (a) - 2 cos (a))

sin^{2} (a) - 2 sin (a) cos (a)

Apply exponent rule:

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

sin^{2} (a) = sin (a) sin (a)

= sin (a) sin (a) - 2 sin (a) cos (a)

Factor out common term

sin (a)

= sin (a) (sin (a) - 2 cos (a))

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

Solving each part separately

sin (a) = 0 or sin (a) - 2 cos (a) = 0

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) - 2 cos (a) = 0 : a = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

cos (a), cos (a) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (a) - 2 = 0

tan (a) - 2 = 0

Move

2

to the right side

tan (a) - 2 = 0

Add

2

to both sides

tan (a) - 2 + 2 = 0 + 2

Simplify

tan (a) = 2

tan (a) = 2

Apply trig inverse properties

tan (a) = 2

General solutions for

tan (a) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

a = arctan (2) + πn

a = arctan (2) + πn

Combine all the solutions

a = 2 πn, a = π + 2 πn, a = arctan (2) + πn

Show solutions in decimal form

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

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tan (x) = \frac{35}{29}

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3 sin (θ) = 1 - sin (θ)

1=sin(90-θ)-0.075cos(90-θ)

1 = sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ)

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- 6 sin (θ) = - 32

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cos (x) = \frac{4 ( \frac{1}{2} ) - 1}{3}

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(a)=2sin(a)cos(a) ?
The general solution for sin^2(a)=2sin(a)cos(a) is a=2pin,a=pi+2pin,a=1.10714…+pin