You can
put this solution on YOUR website!
The answer can be found immediately with a calculator. However
since you are asking this, I am supposing you are asking to show
why without using any numerical calculations. You can use only
information about the signs in the various quadrants and the fact
that an angle is increasing in QI
A = 340° is in QIV and has reference angle 360°-340° = 20°
thus sin(340°) = -sin(20°)
A/2 = 170° is in QII and has reference angle 180°-170° = 10°
thus sin(170°) = sin(10°)
And we know that sin(20°) > sin(10°))
We put question marks for the signs + or - to
include all choices A), B), C), D)
2sin(170°) = ? √(1+sin340°) ? √(1-sin340°)
becomes:
2sin(10°) = ?√(1-sin(20°) ? √(1+sin(20°)
Since √(1+sin(20°) has the larger absolute value it must be
positive. So B) and C) are eliminated.
2sin(10°) = ?√(1-sin(20°) + √(1+sin(20°)
We square both sides:
4sinē(10°) = 1 -sin(20°) ? 2√(1-sin(20°))√(1+sin(20°)) + 1+sin(20°)
Note that the ? did not change in the squaring process.
4sinē(10°) = 2 ? 2√(1-sinē(20°))
2sinē(10°) = 1 ? √cosē(20°)
2sinē(10°) = 1 ? √cosē(20°)
There is an Identity which states:
thus
Therefore the ? must be -
Therefore the only correct choice is D).
Edwin
