# uc3rmw_0403e08

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Hi there.
In this example, we will use
reciprocal and quotient identities
to solve the following problem.
The problem states, given that sine
of t is equal to 3/5 and cosine of t
is equal to negative 4/5, let's find
cosecant of t, secant of t, tangent of t,
and cotangent of t.
First, let's recall what the
reciprocal and quotient identities are.
For the quotient identities,
we have that tangent of t
is equal to sine of t
divided by the cosine of t.
For cotangent of t, we have that it's
the ratio of cosine of t to sine of t.
The reciprocal identities are as follows.
The cosecant of t is equal
to 1 over the sine of t.
Sine of t is equal to 1
over the cosecant of t.
Secant of t is equal to 1 over cosine of t.
Cosine of t is equal to 1 over secant of t.
We have that cotangent of t is equal to
1 over tangent of t, and tangent of t
is equal to 1 over cotangent of t.
The first step to solving this problem
is to use the quotient identity
to find tangent of t.
We're given that sine of t is equal to
3/5 and cosine of t is equal to minus 4/5.
Again, using the quotient
identity, we have that tangent of t
is equal to sine of t over cosine of t.
Plugging in that sine of t is equal to
3/5 and cosine of t is equal to minus 4/5,
we have sine over t over cosine
of t is 3/5 over minus 4/5,
and that simplifies to minus 3/4.
Now we can find cosecant of t,
secant of t, and cotangent of t
using the reciprocal identities.
We have cosecant of t is
equal to 1 over the sine of t.
Where sine of t is equal to 3/5, we have
that this is equal to 1 over 3/5, or 5/3.
For the secant of t, we have that
that's equal to 1 over the cosine of t.
Where cosine of t is minus 4/5,
this is equal to 1 over minus 4/5,
which simplifies to minus 5/4.
Now we can find cotangent of t.
That's equal to 1 over tangent of t.
Tangent of t we found earlier
using the quotient identity.
It's minus 3/4.
So we plug that in.
We have 1 over minus 3/4,
which simplifies to minus 4/3.
Now we found all six of the trigonometric
functions which we obtained starting
from just the values for sine and cosine.
So now we can summarize all six
trigonometric functions in a box.
We have sine of t is equal to 3/5.
Cosine of t is equal to minus 4/5.
Cosecant of t is equal to 5/3.
Secant of t is equal to minus 5/4.
Tangent of t is equal to minus 3/4.
And cotangent of t is equal to minus 4/3.