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Popular >

solvefor θ,3r^2cos^2(θ)+r^2sin^2(θ)<= 1

Solution

solve for

θ, 3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) \leq 1

Solution

False for all θ \in R

Solution steps

3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) \leq 1

Periodicity of

3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) : π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

3 r^{2} cos^{2} (θ), r^{2} sin^{2} (θ)

Periodicity of

3 r^{2} cos^{2} (θ) : π

Periodicity of

cos^{n} (x) = \frac{Periodicity of cos ( x )}{2},

if n is even

Periodicity of

cos (θ) : 2 π

Periodicity of

cos (x)

is

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Periodicity of

r^{2} sin^{2} (θ) : π

Periodicity of

sin^{n} (x) = \frac{Periodicity of sin ( x )}{2},

if n is even

Periodicity of

sin (θ) : 2 π

Periodicity of

sin (x)

is

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Combine periods:

π, π

= π

Factor

3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) : r^{2} (3 cos^{2} (θ) + sin^{2} (θ))

3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ)

Factor out common term

r^{2}

= r^{2} (3 cos^{2} (θ) + sin^{2} (θ))

= No Solution for θ \in R

False for all θ \in R

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