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Exploration 5-3a: Definitions of Sine and Cosine
Date:
Objective: Learn the formal definitions of sine and cosine functions.
1. The figure shows an angle of C =
in standard
position in a uv-coordinate system. Measure
the angle with a protractor. Do you agree that it
370
is
370?
5. The definitions of sine and cosine can be extended to
angles that measure rotation with the aid of the
reference angle. Sketch an angle of 0 = 1250. Then
mark and calculate the reference angle, 0.
V
V
U
6. Use your calculator to find
sm
2. The figure shows a point on the terminal side of (1.
The u- and v-coordinates of the point form a right
triangle whose hypotenuse is the distance from the
origin to the point. Measure the three distances, to
the nearest 0.1 cm.
Adjacent leg, u
Hypotenuse, r =
3. Recall from previous courses that the sine and
cosine of an angle in a right triangle are defined
—
—
opposite leg
hypotenuse
cos
=
COS
=
9
r
ef
=
7. The formal definitions of sine and cosine are
sin
cos
=
Opposite leg, v =
•
9
r
ef
e
vertical coordinate
radius
= horizontal coordinate
radius
=
Calculate sin 125° and cos 125°. How are these
numbers related to the sine and cosine of the
reference angle in Problem 5? How do you explain
that cos 125° is negative?
adjacent leg
hypotenuse
Use the answers in Problem 2 to calculate
sin 37°
cos 37°
4. With your calculator in degree mode, find values of
sin 37° and cos 37°. Do your approximate values in
Problem 3 agree with these precise values?
sin37°
cos
=
370 =
8. State what sign the sine and cosine will have for
angles that terminate in
Quadrant I:
sine
cosine
Quadrant II:
sine
cosine
Quadrant III:
sine
cosine
Quadrant IV:
sine
cosine
9. What did you learn as a result of doing this
exploration that you did not know before?
Precakulus with Trigonomehy: Instructor’s Resource Book
© 2012 Key Curriculum Press
Exploration Masters
123
—r
LA__L...
Problem Set 5-3/Pages 258, 260
Example 3
y
L.j
S in
e
Dote:
53co.c
‘7.
y=cos8
Figure 5-3j
‘5.
y
18.
y
s(e—o°)
y=cos8
I I
-
y=sin8
19.
3-f-cos 2
16.
y
•...ycosQ
y=sin.O
20.
30 Blackhne Masters
4cos(e +CoO0)
Precalcukis with Trigonometry: Instructor’s Resource Book
© 2012 Key Curriculum Press