The-maximum-and-minimum-values-of-a-cos-2-b-sin-2-are-

Question Number 48526 by Pk1167156@gmail.com last updated on 25/Nov/18

The maximum and minimum values

of a \cos 2 θ + b \sin 2 θ are

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Nov/18

a = r s i n α b = r c o s α

a^{2} + b^{2} = r^{2} s o r = \sqrt{a^{2} + b^{2}}

S = a c o s 2 θ + b s i n 2 θ

= r s i n α c o s 2 θ + r c o s α s i n 2 θ

= r s i n (α + 2 θ)

m a x v a l u e o f s i n (α + 2 θ) = 1

m i n v a l u e o f s i n (α + 2 θ) = - 1

s o m a x v a l u e o f a c o s 2 θ + b s i n 2 θ i s

= r \times 1

= \sqrt{a^{2} + b^{2}}

m i n v a l u e o f a c o s 2 θ + b s i n 2 θ

= r \times - 1

= - \sqrt{a^{2} + b^{2}}

Answered by ajfour last updated on 25/Nov/18

a \cos 2 θ + b \sin 2 θ

= \sqrt{a^{2} + b^{2}} \sin (\tan^{- 1} \frac{a}{b} + 2 θ)

s o m a x i m u m v a l u e = \sqrt{a^{2} + b^{2}}

a n d m i n i m u m v a l u e = - \sqrt{a^{2} + b^{2}} .

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