Find the value of sec(θ) if tan(θ)= and sin(θ)>0.
There are two ways to do this problem. We'll do it both ways:
First way. θ is in quadrant II because that
is the only quadrant in which the tangent
is a negative number and the sine is a
positive number.
So we draw the angle θ in quadrant II.
The tangent = = . So
we take the numerator of the tangent as y,
(we take it positive because it goes up from the x-axis).
And we take the denominator of the tangent -19 as x,
(we take it negative because it goes left from the origin).
We use the Pythagorean theorem:
or rē = xē + yē
rē = 19ē +
rē = 361 + 114
rē = 475
r = = =
(r is always taken positive).
And since we know that the secant = = ,
sec(θ) = =
--------------------------------
Second way:
Use the identity:
1 + tanē(θ) = secē(θ)
1 + = secē(θ)
1 + = secē(θ)
+ = secē(θ)
= secē(θ)
= sec(θ)
= sec(θ)
= sec(θ)
= sec(θ)
We have to decide which quadrant θ is in just
as we did using the first method. So the secant
in QII is negative, so the answer is
= sec(θ)
You can do the problem either way.
Edwin