SOLUTION: Hi if sin theta is -4/9 and theta isin the 4th quadrant find the exact value of tan theta and sec theta. could you please show your workings. Thanks

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Question 1117358: Hi
if sin theta is -4/9 and theta isin the 4th quadrant find the exact value of tan theta and sec theta.
could you please show your workings.
Thanks
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52925) About Me  (Show Source):
You can put this solution on YOUR website!
.
You are given  sin(t) = -4%2F9  and "t" is in the 4-th quadrant.


In QIV cos(t) is positive, and therefore


    cos(t) = sqrt%281-sin%5E2%28t%29%29 = sqrt%281-%28-4%2F9%29%5E2%29 = sqrt%281+-+16%2F81%29 = sqrt%28%2881-16%29%2F81%29 = sqrt%2865%2F81%29 = sqrt%2865%29%2F9.


Then tan(t) = sin%28t%29%2Fcos%28t%29 = %28%28-4%2F9%29%29%2F%28%28sqrt%2865%29%2F9%29%29 = -4%2Fsqrt%2865%29 = rationalize the denominator = %28-4%2Asqrt%2865%29%29%2F65.


Next  sec(t) = 1%2Fcos%28t%29 = 1%2F%28%28sqrt%2865%29%2F9%29%29 = 9%2Fsqrt%2865%29 = rationalize the denominator = %289%2Asqrt%2865%29%29%2F65.


Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
sin(theta) = -4/9

this means opposite side to the angle is -4 and hypotenuse is 9.

this forms a right triangle on the unit circle where x is the length of the adjacent side
and y is the length of the opposite side and h is the length of the hypotenuse.

sin(theta) = y/h

cos(theta) = x/h

tan(theta) = y/x

since it's a right triangle, then x^2 + y^2 = h^2.

this gets you x^2 + (-4)^2 = 9^2

this results in x^2 + 16 = 81

solve for x^2 to get x^2 = 81 - 16 = 65

solve for x to get x = sqrt(65)

you now have:

x = sqrt(65) = adjacent side
y = -4 = opposite side
h = 9 = hypotenuse

tan(theta) = y/x = -4/sqrt(65)

sec(theta) = 1/cos(theta) = 1/(x/h) = h/x = 9/sqrt(65)