Question

2 Suppose √( 9 − 8 cos 40◦) = a + b sec 40◦ , where a and b are rational numbers. Then |a + b| equals (A) 1 2 (B) 3 2 (C) 2 (D) 3 ​

Answer 1

√√s40°)
={cos40°√(9-8cos40°)}/cos40°
=√(9cos²40°-8cos³40°)/cos40° --------------------------(1)
Now we know,
cos3A=4cos³A-3cosA
or, 4cos³A=cos3A+3cosA
or, 8cos³A=2cos3A+6cosA
or, 8cos³40°=2cos(3×40°)+6cos40°
or, 8cos³40°=2cos120°+6cos40°
or, 8cos³40°=-2cos60°+6cos40°
[∵, cos120°=cos{(90°×2)-60°}=-cos60°]
or, 8cos³40°=-1+6cos40° [∵, cos60°=1/2]
Putting in (1) we get,
√(9-8cos40°)=√(9cos²40°-8cos³40°)/cos40°
=√(9cos²40°+1-6cos40°)/cos40°
=√{(3cos40°)²-2.3cos40°.1+1²}/cos40°
=√(3cos40°-1)²/cos40°
=|3cos40°-1|/cos40°
=|3-1.sec40°|
∴, a+bsec40°=|3+(-1)sec40°|
Equating coefficients from both sides, a=3, b=-1
∴, |a+b|=|3+(-1)|=2 Ans.

Answer 2

Answer:

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