Theory question 101
TrigonometryIf tan x = 1/3, where 180° < x < 270° evaluate (sin 2x − cos x)/(2 tan x + sin 2x), leaving the answer in surd form (radicals)
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Model answer
x is in the third quadrant, so sin x and cos x are both negative: sin x = −1/√10, cos x = −3/√10.
sin 2x = 2 sin x cos x = 6/10 = 3/5.
Numerator: 3/5 − (−3/√10) = 3/5 + 3/√10. Denominator: 2/3 + 3/5 = 19/15.
Value = (15/19)(3/5 + 3/√10) = 9/19 + 45/(19√10) = (18 + 9√10)/38 = 9(2 + √10)/38.
Explanation
Use a right triangle with sides 1, 3, √10 and take the signs for the third quadrant.