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Question

Find the values of the other five trigonometric functions in each of the following:

(i) cot x=125, x in quadrant III

(ii) cos x=-12, x in quadrant II

(iii) tan x=34, x in quadrant III

(iv) sin x=35, x in quadrant I

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Solution

i We have:cotx = 125 and x are in the third quadrant. In the third quadrant, tanx and cotx are positiveAnd, sinx, cosx , secx and cosecx are negative.
tanx = 1cotx = 1125=512cosecx = -1 + cot2x = -1 +1252 = -135sinx = 1cosecx = 1- 135 = -513cosx = -1 - sin2x = -1 - -5132 = -1213And, secx = 1cosx = 1-1213 = -1312
ii We have:cosx =-12 and x are in the second quadrant.In the second quadrant, sinx, and cosecx are positive.And, tanx, cotx , cosx and secx are negative.

sinx = 1 - cos2x = 1 - -122 = 32tanx = sinxcosx = 32-12=-3
cotx= 1tanx = 1-3=-13 secx = 1cosx = 1-12 = -2cosecx = 1sinx = 132 = 23
iii We have:tanx = 34 and x are in the third quadrant.In the third quadrant, tanx, and cotx are positive.And, sinx, cosx , secx and cosecx are negative.
cotx = 1tanx = 134=43secx = -1 +tan2x = -1 +342 = - 54cosx = 1secx = 1- 54 = -45sinx = -1 - cos2x = -1 - -452 = -35cosecx = 1sinx = 1- 35 = -53
iv We have:sinx =35 and x are in the first quadrant.In the first quadrant, all six T-ratios are positive.
cosx = 1 - sin2x = 1 - 352 = 45tanx = sinxcosx = 3545=34cotx = 1tanx = 134=43secx = 1cosx = 145 = 54cosecx = 1sinx = 135 = 53

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