Expert: Ahmed Salami - 1/31/2010

Question
In each case, find sin(α), cos(α), tan(α), csc(α), sec(α), and cot(α).
sin(2α) = -15/17 and 180° < 2α < 270°

Answer
Hi Claire,
sin(2α) = -15/17
cos(2α) = √[1 - sin²(2α)]
       = √[1 - (-15/17)²]
       = √[1 - (225/289)]
       = √(64/289)
       = ±8/17
but 180° < 2α < 270° (which means that cos(2α) is negative) and so
cos(2α) = -8/17
Now,
cos(2α) = 2cos²α - 1
-8/17 = 2cos²α - 1
2cos²α = 9/17
cos²α = 9/34
cosα = √(9/34)
    = ±(3√34)/34
but 90° < α < 135° and cosα is negative
cosα = -(3√34)/34

sin(2α) = 2sinα.cosα
sinα = sin(2α)/2cosα
    = (-15/17)/[-(3√34)/34]
    = (5√34)/34

tanα = sinα/cosα
    = [(5√34)/34] / [(-3√34)/34]
    = -5/3

cscα = 1/sinα
    = 1/[(5√34)/34]
    = √34/5

secα = 1/cosα
    = 1/[(-3√34)/34]
    = -√34/3

cotα = 1/tanα
    = 1/(-5/3)
    = -3/5

Regards