Question

If a = cos 3 & b = sin 8 then
A) \[a > 0,b > 0\]
B) \[ab < 0\]
C) \[a > b\]
D) \[ab > 0\]

Answer 1

Hint: 3 and 8 given in cosine and sine respectively are clearly in radian not in degrees and also note that 1 radian = \[{57^ \circ }\] use this to change the radians in degrees and then convert them all to there principal values so that finally we can judge our answer.
Complete Step by Step Solution:
Given is \[a = \cos 3\& b = \sin 8\]
We know that \[{1^c} = {57^ \circ }\]
Therefore, \[{3^c} = 171\& {8^c} = 456\]
Now let us substitute the values of radians, we will get it as
\[\begin{array}{l}
a = \cos 3 = \cos {171^ \circ }\\
b = \sin 8 = \sin {456^ \circ }
\end{array}\]
Now we can write
\[\begin{array}{l}
a = \cos {(90 + 81)^ \circ } = - \sin {81^ \circ }\\
b = \sin {(360 + 96)^ \circ } = \sin {96^ \circ }
\end{array}\]
Now \[\sin {96^ \circ }\] is in second quadrant and we know that sine is positive in second quadrant therefore here \[a < 0\& b > 0\]
\[\therefore ab < 0\]
Which means option B is correct.

Note: Any number to the power c basically denotes radian like in this case it was \[{3^c},{8^c},{1^c}\] which means 3 radian, 8 radian and 1 radian respectively. Also option C is incorrect because as \[a < 0\& b > 0\] means \[b > a\]