Download MTH 112, Class Notes, Date: Section 1.3, Trigonometric Functions 1

MTH 112, Class Notes, Date:
Section 1.3, Trigonometric Functions
1. The study of trigonometry covers the six
functions, we start with an angle in
having coordinates
on the
gle
. (The point P must not be the
from P to the
a
, having vertices at
the distance
from
to the origin,
formula.
,
. To define these
, and choose any point
of the anof the angle.) A
at point
determines
, and
. We find
, using the distance
r=
2. The six trig functions of an angle θ are
,
, and
,
,
,
.
3. Trigonometric functions: Let (x, y) be a point other than the origin on the terminal
side of an angle θ in standard position. The distance from the point to the origin is
r=
. The six trig functions of θ are defined as follows.
sine θ
=
=
cosecant θ
=
=
cosine θ
=
=
secant θ
=
=
tangent θ =
=
cotangent θ =
=
Example 1. The terminal side of an angle θ in standard position passes through the point
. Find the values of the six trigonometric functions of angle θ.
Example 2. The terminal side of an angle θ in standard position passes through the point
. Find the values of the six trigonometric functions of angle θ.
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MTH 112, Class Notes, Section 1.3, Trigonometric Functions
4. We can find the six trigonometric functions using
on the
side of an angle.
point other than the
Example. The angle, θ, has two distinct points on its terminal side,
and
. Let
be the length of the hypotenuse of triangle
,
and let
be the length of the hypotenuse
. Since corresponding
sides of similar triangles are
,
So
is the
no matter which point is used to find it. A similar
reasoning holds for the other five trig functions.
5. We can also find the trig function values of an angle if we know the
coinciding with the
6. Recall:
.
is a line that passes through the origin.
7. If we restrict x to have only
(or
the graph a
with endpoint at the
side of an angle θ in
position.
) values, we obtain as
. This can serve as a
Example 3. Find the six trigonometric function values of the angle θ in standard position,
if the terminal side of θ is defined by
,
.
8. Recall. Slope-intercept form:
9. In general, it is true that m =
,m=
.
.
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MTH 112, Class Notes, Section 1.3, Trigonometric Functions
10. Note. The trig function values we found in Examples 1-3 are
. If we
were to use a calculator to
these values, the decimal results would
be acceptable if
values were required.
Quadrantal Angles:
11. If the terminal side of an angle in standard position lies along the
, any
point on this terminal side has x-coordinate
. Similarly, an angle with terminal
side on the
has y-coordinate
for any point on the terminal side.
12. Since the values of x and y appear in the
a fraction is
if its denominator is
for quadrantal angles.
13. Because
the point
of some trig functions, and since
, some trig function values are
point on the terminal side can be used, it is convenient to choose
, with
.
14. To find the function values of a quadrantal angle, determine the position of the
side, choose the point
that lies on this terminal side, and then
involving x, y, and r.
use the
Example 4. Find the values of the six trigonometric functions for each angle.
(a) an angle of
(b) an angle θ in standard position with terminal side through
.
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MTH 112, Class Notes, Section 1.3, Trigonometric Functions
15. Summary: If the terminal side of a quadrantal angle lies along the
and
functions are undefined. If it lies along the
then the
and
functions are undefined.
, then the
16. You can find a table of the trig function values for the most commonly used quadrantal
angles on page 27.
17. You can find these values with a calculator, but make sure the calculator is set in
mode. One of the most common errors involving calculator in trig occurs
when the calculator is set for
measure, rather than
measure.
(We will discuss radians in Chapter 3.)
Homework:
,