Find the exact values of the remaining trigonometric functions

Question

Find the exact values of the remaining trigonometric functions of θ from the given information.

tan θ = - 5/12, θ in quadrant II

cot θ = - 12/5, θ in quadrant IV

Answer 1

tanT =o/a= -5/12 in quadrant 2, cotT =a/o= -12/5 = -1.4 in quadrant 4

T is an angle in quadrant 2 or 4

it's a Pythagorean triple for the sides 5-12-13, 13=hypotenuse=h

12=adjacent side=a, 5=opposite side=o

13^2 = 12^2 +5^2 = 144+25 = 169 = 13x13

sinT =o/h= -5/13 cscT = 13/-5=-13/5=-1.6

cosT =a/h =12/13, secT = 13/12

Answer 2

You have to use a combination of the trig identities and knowledge of which functions are positive in which quadrant.

For example:

we know that tan^2(x) + 1 = sec^2(x). Knowing tan x lets you know sec (and cos) and you can find the rest from the other identities. You need to know the sign of each function (from the quadrant information) because the identities involve squaring so taking sqrt(sec^2(x)) for example only tells you |sec(x)|.

Answer 3

A triangle with sides 5, 12 and 13 is a right triangle.

The sine is 5/13 and the cos is 12/13.

You can now figure the others.

You should remember the mnemonic AllSinTanCos which tells which functions are + in which quadrant.