Now I'll finish my picture by adding
the length of the hypotenuse to my right triangle:

And that's all I need for finding my
ratios! To find my answers, I can just read the number from my picture:

Determine the quadrant
in which lies the terminal side of the angle theta, given that tan(θ)
< 0 and sin(θ)
< 0.

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For this exercise, I need to consider
the x-
and y-values
in the various quadrants, in the context of the trig ratios. I don't
need to find any actual values; I only need to work with the signs and
what I know about the ratios and the quadrants.

The tangent ratio is y/x,
so the tangent will be negative when x
and y
have opposite signs. This occurs in the second quadrant (where x
is negative but y
is positive) and in the fourth quadrant (where x
is positive but y
is negative). So the sign on the tangent tells me that the end of the
angle is in QII
or in QIV.

The sine ratio is y/r,
and the hypotenuse r
is always positive. So the sine will be negative when y
is negative, which happens in the third and fourth quadrants.

So the tangent is negative in QII
and QIV,
and the sine is negative in QIII
and QIV.
The overlap between the two solutions is QIV,
so:

The terminal side of the angle θ
lies somewhere in
QIV.

The thought process for the exercise above
leads to a rule for remembering the signs on
the trig ratios in each of the quadrants. In the first quadrant, all the
values (x,
y,
and r) are positive, so All the trig ratios are positive. In the second
quadrant, the x-values
are negative, so x/r
and y/x
are negative; only y/r
is positive, so only the Sine is positive in QII.
In the third quadrant, each of x and
y is
negative, so x/r
and y/r
are negative; only y/x
is positive, so only the Tangent is positive in QIII.
In the fourth quadrant, the y-values
are negative, so y/r
and y/x
are negative; only x/r
is positive, so only the Cosine is positive in QIV.

You're probably wondering why I capitalized
the trig ratios and the word "All" in the preceding paragraph.
I did that to explain this picture:

The letters in the quadrants stand
for the initials of the trig ratios which are positive in
that quadrant.

Some people remember the letters using
the word "ACTS", but that's the reverse of normal (anti-clockwise)
trigonometric order. Others remember the letters with the word "CAST",
which is the normal rotational order but doesn't start in the usual (first-quadrant)
starting place. To start in the usual spot and rotate in the usual direction,
still others use the mnemonic "All Students Take Calculus" (which
is a bit ironic when you're in a trig class). Use whichever method works
best for you.

Find the values of
the remaining trigometric ratios, given that cos(θ)
= –8/17 and
theta lies in QIII.

From the sign on the cosine value, I
only know that the angle is in QII
or QIII.
That's why they had to give me that additional specification: so I'd
know which of those quadrants I'm in.

I'll start my work by drawing a
picture of what I know so far: