How do you find the values of the trigonometric functions of θ from the information given cot θ = 1/4, sin θ < 0?

1 Answer
Noah G
Dec 2, 2016

First of all, #cottheta = 1/tantheta = 1/(1/4) = 4#.

The problem says that sine is negative and we can see above that tangent is negative. Using the C-A-S-T sign rule, we determine that #theta# is in Quadrant 3.

We know that #tantheta = "opposite"/"adjacent"#, so our opposite side measures #-4# units and our adjacent side measures #-1# unit (because #(x, y) = (-, -)# in quadrant 3).

We use pythagorean theorem to find the hypotenuse.

#(-4)^2 + (-1)^2 = h^2#

#16 + 1 =h^2#

#h =+-sqrt( 17#

However, the hypotenuse can never be negative, so we only keep #sqrt(17)#.

We can now fill in all four of the other ratios:

#sintheta = "opposite"/"hypotenuse" = -4/sqrt(17)#

#csctheta = 1/sintheta = "hypotenuse"/"opposite" = -sqrt(17)/4#

#costheta = "adjacent"/"hypotenuse" = -1/sqrt(17)#

#sectheta = 1/costheta = "hypotenuse"/"adjacent" = -sqrt(17)#

Hopefully this helps!

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