Question

find the minimum value of sin squared theta + cos to the power 4 theta​

Answer 1

Answer:

Step-by-step explanation:

To find the maxima and minima of this function differentiate it and equate it to zero (t = theta):

F(t) = Sin^4(t) + Cos^4(t), therefore,

F’(t) = 4.Sin^3(t)Cos(t) - 4.Cos^3(t)Sin(t) = 0

(the 4s cancel, then factorise):

Sin(t)Cos(t)[Sin^2(t) - Cos^2(t)] = 0

There are turning points when t = 0, and when t = 90. The other turning points occur when

Sin^2(t) = Cos^(t) or when t = 45

Cos(45) = Sin(45) = 1/[sqrt(2)], substitute t = 45 in F(t), which yields

1/4 + 1/4 = 1/2

Maximum values of F(t) is 1 and occurs at t = 90 and t = 0 with minimum at t = 45 (for the principal range).