What is the general solution for cos^2(x)+sin^2(2x)=0 ?

The general solution for cos^2(x)+sin^2(2x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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cos^2(x)+sin^2(2x)=0

Solution

cos^{2} (x) + sin^{2} (2 x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + sin^{2} (2 x) = 0

Rewrite using trig identities

cos^{2} (x) + sin^{2} (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Simplify

cos^{2} (x) + (2 sin (x) cos (x))^{2} : cos^{2} (x) + 4 sin^{2} (x) cos^{2} (x)

cos^{2} (x) + (2 sin (x) cos (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

= cos^{2} (x) + 4 sin^{2} (x) cos^{2} (x)

= cos^{2} (x) + 4 sin^{2} (x) cos^{2} (x)

cos^{2} (x) + 4 cos^{2} (x) sin^{2} (x) = 0

Factor

cos^{2} (x) + 4 cos^{2} (x) sin^{2} (x) : cos^{2} (x) (4 sin^{2} (x) + 1)

cos^{2} (x) + 4 cos^{2} (x) sin^{2} (x)

Factor out common term

cos^{2} (x)

= cos^{2} (x) (1 + 4 sin^{2} (x))

cos^{2} (x) (4 sin^{2} (x) + 1) = 0

Solving each part separately

cos^{2} (x) = 0 or 4 sin^{2} (x) + 1 = 0

cos^{2} (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

4 sin^{2} (x) + 1 = 0 :

No Solution

4 sin^{2} (x) + 1 = 0

Solve by substitution

4 sin^{2} (x) + 1 = 0

Let:

sin (x) = u

4 u^{2} + 1 = 0

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

4 u^{2} + 1 = 0

Move

1

to the right side

4 u^{2} + 1 = 0

Subtract

1

from both sides

4 u^{2} + 1 - 1 = 0 - 1

Simplify

4 u^{2} = - 1

4 u^{2} = - 1

Divide both sides by

4

4 u^{2} = - 1

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{- 1}{4}

Simplify

u^{2} = - \frac{1}{4}

u^{2} = - \frac{1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - \frac{1}{4}, u = - - \frac{1}{4}

Simplify

- \frac{1}{4} : i \frac{1}{2}

- \frac{1}{4}

Apply radical rule:

- a = - 1 a

- \frac{1}{4} = - 1 \frac{1}{4}

= - 1 \frac{1}{4}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{4} = \frac{1}{4}

= i \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= i \frac{1}{2}

Apply rule

1 = 1

= i \frac{1}{2}

Rewrite

i \frac{1}{2}

in standard complex form:

\frac{1}{2} i

i \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 i}{2}

Multiply:

1 i = i

= \frac{i}{2}

= \frac{1}{2} i

Simplify

- - \frac{1}{4} : - i \frac{1}{2}

- - \frac{1}{4}

Simplify

- \frac{1}{4} : i \frac{1}{2}

- \frac{1}{4}

Apply radical rule:

- a = - 1 a

- \frac{1}{4} = - 1 \frac{1}{4}

= - 1 \frac{1}{4}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{4} = \frac{1}{4}

= i \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= i \frac{1}{2}

= - i \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2} i

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

Substitute back

u = sin (x)

sin (x) = i \frac{1}{2}, sin (x) = - i \frac{1}{2}

sin (x) = i \frac{1}{2}, sin (x) = - i \frac{1}{2}

sin (x) = i \frac{1}{2} :

No Solution

sin (x) = i \frac{1}{2}

No Solution

sin (x) = - i \frac{1}{2} :

No Solution

sin (x) = - i \frac{1}{2}

No Solution

Combine all the solutions

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Popular Examples

2cos^2(x)+3sin^2(x)=2

2 cos^{2} (x) + 3 sin^{2} (x) = 2

42-25cos(x)=31

42 - 25 cos (x) = 31

2sec^2(x)=5tan(x)

2 sec^{2} (x) = 5 tan (x)

cos^2(2x)-3sin^2(x)-1=0

cos^{2} (2 x) - 3 sin^{2} (x) - 1 = 0

sin(-t)= 3/10

sin (- t) = \frac{3}{10}

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)+sin^2(2x)=0 ?
The general solution for cos^2(x)+sin^2(2x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin