If x = a \sin^2 \theta\sin\theta , y = a \sin^2 \cos \theta , then which one of the following expressions is independent of θ? 1. (x^2+y^2)^3/(x^2y^2) 2. \sqrt{(x^2+y^2)}/\sqrt^3 {xy} Select the correct answer using the codes given below:

Correct Answer is : both 1 and 2

Trigonometry Practice Test 2

Question : 6 of 22

Marks: +1, -0

x = a \sin^2 θ\sinθ

y = a \sin^2 \cos θ

, then which one of the following expressions is independent of θ?
1.

(x^2+y^2)^3/(x^2y^2)

√{(x^2+y^2)}/√^3 {xy}

Select the correct answer using the codes given below:

Solution:

Given that

x = a \sin^2 θ\sinθ

y = a \sin^2 \cos θ

(x^2+y^2)^3/(x^2y^2)

={(a^2\cos^4θ\sin^2θ+a^2\sin^4θ\cos^2θ)^3}/{a^4\cos^4θ\sin^2θ\sin^4θ\cos^2θ}

={a^6\sin^6θ\cos^6θ}/{a^4\sin^6θ\cos^6θ}=a^2

This shows that the expression

{(x^2+y^2)^3}/{x^2y^2}

is independent of θ.
2.

√{(x^2+y^2)}/√^3 {xy}

=√{a^2\cos^4θ\sin^2θ+a^2\sin^4θ\cos^2θ}/{√^3{\sin^3θ\cos^3θ}}

={a\sinθ\cosθ}/{a^{2/3}\sinθ\cosθ}=a^{1/3}

This shows that the expression

√{(x^2+y^2)}/√^3 {xy}

is also independent of θ.
So, option (c) is correct.

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