find exact value of tan theta= -1/3, sin theta > 0

Answer 1

tanx =-1/3

x = -18.4 degrees, plus 180n where n= any integer

that's the 4th quadrant or 2nd quadrant

but sinx >0, putting x in either the 1st or 2nd quadrant.

that means x must be in the 2nd quadrant only, and

x = 180-18.4 = 161.6 + 360n where n = any integer

still that's not an "exact" value, but a rounded value to one decimal place

exact value is x= tan^-1(-1/3)

= the inverse tangent of -1/3

= 161.5650512 rounded to 7 decimal places or 10 digits which is a little more "exact" English instructors may disagree, but our Founding Father's adopted a preamble to the Constitution with the words in order to form a "more perfect" union..."more perfect" is similar to "a little more exact." Of course, once you have a perfect or exact solution, you can't get "more perfect" or "more exact," but new dictionaries today have included "figuratively" as a possible definition for "literally." 161.5650512 is literally exact if you consider it as figuratively exact or more exact than more rounded solutions. Usually though, for trig angles, when they ask for "exact" they tend to mean leave it in pi radians rather than the rounded degree equivalent. but for this example, the pi version would also be inexact