what is the minimum value of cos^4 theta+sin^2 theta - Maths - Trigonometric Functions

Question

what is the minimum value of cos^4 theta+sin^2 theta ?

Ajanta Trivedi · Accepted Answer

let the given function is:

$f (θ) = \cos^{4} θ + \sin^{2} θ = (\cos^{2} θ)^{2} + \sin^{2} θ = (1 - \sin^{2} θ)^{2} + \sin^{2} θ = 1 + \sin^{4} θ - 2 \sin^{2} θ + \sin^{2} θ = 1 + \sin^{4} θ - \sin^{2} θ$

$= 1 - \sin^{2} θ (1 - \sin^{2} θ) = 1 - \sin^{2} θ \cos^{2} θ = 1 - \frac{1}{4} (2 \sin θ \cos θ)^{2} = 1 - \frac{1}{4} \sin^{2} 2 θ$

since maximum value of $\sin^{2} 2 θ$ is 1.

thus

$f (θ) \geq 1 - \frac{1}{4} \cos^{4} θ + \sin^{2} θ \geq \frac{3}{4}$

thus minimum value of $\cos^{4} θ + \sin^{2} θ$ is 3/4.

hope this helps you.

cheers!!