Latest Others Aptitude Question SOLUTION: Which one of the two is greater, sin(cos a) or cos(sin a)?

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Which one of the two is greater,
sin(cos a) or cos(sin a)?

Asked In Others jyotish r (11 years ago)

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chaitanya  (11 years ago)

We will make use of the following trigonometric identities:

sin A = cos(pi/2 − A)-------------------------------------(1)
cos A − cos B = −2 sin(1/2(A + B)) sin(1/2(A − B))----------(2)
A cos x + B sin x = R cos (x − c)-----------------------(3)
where R = (A2 + B2), and c = arctan(B/A)

Applying the above, we obtain

cos(sin x) − sin(cos x) = cos(sin x) − cos(pi/2 − cos x) from (1).
= −2 sin[1/2(sin x−cos x +pi/2)]sin[1/2(sin x + cos x − /2)]from (2).
= −2 sin(−1/2(2)^1/2 cos(x +pi/4)+pi/4)
X
sin(1/2(2)^1/2cos(x − pi/4)− /4) from (3).

Since pi/4 > 1/2 (2)^1/2, 0 < −1/2 (2)^1/2cos(x +pi/4) + pi/4 0, for all x.

Similarly, −pi/2 < 1/2 (2)^1/2cos(x −pi/4)− pi/4 < 0, for all x.

Hence sin(1/2 (2)^1/2cos(x −pi/4)−pi/4) < 0, for all x.

It follows that −2 sin(−1/2cos(x + pi/4) +pi/4) sin(1/2cos(x −pi/4)−pi/4) > 0.

Therefore cos(sin x) > sin(cos x) for all real x.

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IQBAL MEHDI  (11 years ago)

cos(sin a) > sin(cos a).
sol:-
let a=180(degree)
now,
cos(sin 180')=cos(0)=1

sin(cos 180')=sin(-1)=-0.84147

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Sneha paliwal  (11 years ago)

ans cos(sin a) is greater than sin(cos a) .because when sin a=0 then cos(sin a)=1
and when cos a=0 then also sin(cos a)=0.

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