SOLUTION: Hi can someone help me with this problem please? THANK YOU State the exact value of t,given cot t=-root3 and t [0,2pi) terminates QII Select one

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Question 1077637: Hi can someone help me with this problem please? THANK YOU
State the exact value of t,given cot t=-root3 and t [0,2pi) terminates QII
Select one
a.5pi/6
B.7pi/6
C. pi/3
D. 2pi/3
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You do not need to memorize anything to figure out
the trigonometric ratios of a 30-60-90 right triangle
(one with acute angles measuring pi%2F6=30%5Eo and pi%2F3=60%5Eo ).
That triangle is half of an equilateral triangle.
If the length of the hypotenuse is 1 ,
the short leg's length is 1%2F2 ,

and the Pythagorean theorem lets you figure out
the length of the other leg as h=sqrt%283%29%2F2 .
So, for the pi%2F6=30%5Eo , we have the trigonometric ratios
sin%28pi%2F6%29=1%2F2 , cos%28pi%2F6%29=sqrt%283%29%2F2 , and tan%28pi%2F6%29=%281%2F2%29%2F%28sqrt%283%29%2F2%29=1%2Fsqrt%283%29 .
Although cotangent is not as popular as sine, cosine, and tangent,
it is defined as cot%28theta%29=1%2Ftan%28theta%29,
so cot%28pi%2F6%29=sqrt%283%29 .
You know how angles in other quadrants relate to
symmetrical acute reference angles in quadrant I.

In Quadrant II (QII for short), we have pi-pi%2F6=5pi%2F6=5pi%2F6
as the symmetrical angle to reference angle pi%2F6=5pi%2F6 .
The absolute values of the trigonometric functions for those two angles are the same,
but some signs change:
sin%285pi%2F6%29=sin%28pi%2F6%29=1%2F2 , cos%285pi%2F6%29=-cos%28pi%2F6%29=-sqrt%283%29%2F2 ,
tan%285pi%2F6%29=-tan%28pi%2F6%29=-1%2Fsqrt%283%29 , and cot%285pi%2F6%29=-cot%28pi%2F6%29=-sqrt%283%29

NOTE:
To be able to say that you know trigonometry,
you should be able to figure out (not memorize without understanding)
the trigonometric functions for certain key angles.
You should start with 0=0%5Eo and pi%2F2=90%5Eo ,
based on the unit circle definitions,
because you cannot use right triangle trigonometric ratios for those angles.
From there, you should also know the trigonometric functions' values
for all angles that are multiples of pi%2F2=90%5Eo .
Beyond that, there are key angles between 0=0%5Eo and pi%2F2=90%5Eo
whose trigonometric ratios you should be able to figure out.
Consider a square of side length 1 , split in half by a diagonal.
The two halves are isosceles right triangles,
so both acute angles measure 90%5Eo%2F2=45%5Eo=pi%2F4 ,
and the length, x, of the legs of that right triangle,
can be figured out using the Pythagorean theorem.
Since 2x%5E2=x%5E2%2Bx%5E2=1%5E2=1 , x%5E2=1%2F2 ,
and x=sqrt%281%2F2%29=1%2Fsqrt%282%29=sqrt%282%29%2F2 .
No memorization needed.
For the 30%5Eo=pi%2F6 and 60%5Eo=pi%2F3 ,
consider an equilateral triangle with side length 1 (as shown above)
split into two congruent right triangles by one of its altitudes. .