Download Geo-Trig Rediscovery (WP)

Physics
Circles:
Geometry/Trigonometry
Rediscovery
Draw on the diagrams on the next piece of paper
1 . a . Draw a radius onto circle 1 on the last page. Now take a ruler and
lay it against the circle so that the edge just touches ONLY the point
on the circle where the radius does. Finally, measure the angle
between the radius and the ruler. What is it?
°
b . Is there a way that you can you make the ruler edge touch only o n e
point on the circle, and have the angle between the ruler and the
radius to that point be something other than what you measured in 1a?
c . How would you define the tangent of a circle?
2 . a . Now take a piece of string and cut it to the length of the radius of
circle 2-3. Put one end of the string on the perimeter of the circle
where the radius touches, lay out the string on the perimeter of
the circle, and make a mark on the perimeter where the other end
of the string is. Now draw a radius to this second mark. What is
the angle between the two radii you have drawn?
°
b . How many of these strings does it take to cover
the whole perimeter of the circle?
c . Write an equation to link the length of a
circle’s radius (R) to its circumference
(C)
C =
3 . a . Angles can be measured in a few ways. Two of them are degrees and radians. The
angle you just measured in degrees in 2a is one radian. How many radians fit into a
full circle?
What is your reasoning?
b . Give a definition for a 1 radian angle
c . Using your definition, c a l c u l a t e how many
degrees a radian should be (show your setup).
°
3 . a . Choose an arc of arbitrary length on circle 2-3 (make marks on the perimeter to
define it). Without using a protractor, tell a method for measuring approximately
in radians, what the a n g l e this arc defines is . Now, use the method you just
suggested to actually measure the angle this arc defines.
radians
b . Give an equation to tell how you can calculate an angle measurement (θ) of a
section of a circle in radians, if you know the arc length (call it “S”) and the
length of the radius (R).
c . Now use a protractor to to measure the angle in d egr ees of the arc in 3a . Calculate
what this angle should be in radians. Do your measurements and calculations below.
°
=
radians
P o l y g o n s : Use the diagrams on next piece of paper
1. If you measure all the angles in a triangle, and then add
them together, how many degrees is the total?
°
2 . Look at the 4-sided figure. Measure the 4 angles and add them. What is the total?
A
°
B
°
C
What is the total number of degrees if
you sum up the angles in a rectangle?
°
D
°
°
Total
a square?
°
°
3 . Measure all the angles in the other figures and total them:
5-sided:
A
°
6-sided:
A
°
B
B
°
°
C
C
°
°
D
D
°
°
E
°
E
°
Total
F
°
Total
°
°
4. Look at your data. Can you come up with an equation to relate the number of sides for a
figure (N) to the total number of degrees ( θ t o t a l) when you add all angles together ?
Write it below.
θt o t a l =
Trigonometry
1 . For the triangle shown, give the formula for sin θ , cos θ and tan θ for the angle θ
terms of a, b and c.
sin θ =
cos θ =
tan θ =
2 . Put in clusters (circle each cluster) all of the angles that are the same in the
drawing, if line X is parallel to line Z.
Use one angle around line X and one angle around line Z. Give all pairs of angles,
whose sum is 180°.
in
Geometry/Trigonometry
Lab
Illustrations
Circles:
1
2 and 3
Polygons:
2
B
C
A
C
3B
D
D
B
A
B
E
3A
A
C
F
E
D
X
Trigonometry:
A
1
2
c
a
θ
b
C
B
D
Z
E
F
G H
Physics
Circles:
Geometry/Trigonometry
Rediscovery
Draw on the diagrams on page 9
1 . a . Draw a radius onto circle 1 on the last page. Now take a ruler and
lay it against the circle so that the edge just touches ONLY the point
on the circle where the radius does. Finally, measure the angle
between the radius and the ruler. What is it?
90°
b . Is there a way that you can you make the ruler edge touch only o n e
point on the circle, and have the angle between the ruler and the
radius to that point be something other than what you measured in 1a?
No
c . How would you define the tangent of a circle?
A line drawn perpendicular to any one radius.
2 . a . Now take a piece of string and cut it to the length of the radius of
circle 2-3. Put one end of the string on the perimeter of the circle
where the radius touches, lay out the string on the perimeter of
the circle, and make a mark on the perimeter where the other end
of the string is. Now draw a radius to this second mark. What is
the angle between the two radii you have drawn?
57.3
b . How many of these strings does it take to cover
the whole perimeter of the circle?
c . Write an equation to link the length of a
circle’s radius (R) to its circumference
3 . a . Angles can be
angle you just
full circle?
2
can fit 2πR/R =
(C)
>6 = 2π
C = 2πR
measured in a few ways. Two of them are degrees and radians. The
measured in degrees in 2a is one radian. How many radians fit into a
What is your reasoning?
π
Since the circumference is 2πR, You
2π pie wedges into a full circle.
b . Give a definition for a 1 radian angle
length
angle subtended by an arc of 1 radius
c . Using your definition, c a l c u l a t e how many
degrees a radian should be (show your setup).
360/2π
=
57.3°
3 . a . Choose an arc of arbitrary length on circle 2-3 (make marks on the perimeter to
define it). Without using a protractor, tell a method for measuring approximately
in radians, what the a n g l e this arc defines is . Now, use the method you just
suggested to actually measure the angle this arc defines.
e.g. arclength = 1 1/3 times the radius. If radius is 2.5 cm, then arclength =
1.33 * 2.5 cm = 3.325 cm. Angle = 1.33 * 1 radian =
1.33 radians
b . Give an equation to tell how you can calculate an angle measurement (θ) of a
section of a circle in radians, if you know the arc length (call it “S”) and the
length of the radius (R).
Ø = S/R
c . Now use a protractor to to measure the angle in d egr ees of the arc in 3a . Calculate
what this angle should be in radians. Do your measurements and calculations below.
S = 3.325 cm
R = 2.50 cm
76.2 °
Ø = 3.325 cm/2.50 cm = 1.33 radians
1.33 radians * 57.3 °/radian = 76.2°
1.33
radians
P o l y g o n s : Use the diagrams on page 9
1. If you measure all the angles in a triangle, and then add
them together, how many degrees is the total?
180°
2 . Look at the 4-sided figure. Measure the 4 angles and add them. What is the total?
A
°
B
°
C
°
D
What is the total number of degrees if
you sum up the angles in a rectangle? 3 6 0 °
°
a
Total 360 °
square?3 6 0 °
3 . Measure all the angles in the other figures and total them:
5-sided:
A
°
6-sided:
A
°
B
B
°
°
C
C
°
°
D
D
°
°
E
°
E
°
Total540°
F
°
Total720°
4. Look at your data. Can you come up with an equation to relate the number of sides for a
figure (N) to the total number of degrees ( θ t o t a l) when you add all angles together ?
Write it below.
θt o t a l = (N-2) * 180
Trigonometry
1 . For the triangle shown, give the formula for sin θ , cos θ and tan θ for the angle θ
terms of a, b and c.
sin θ = a / c
cos θ = b/c
tan θ = a/b
2 . Put in clusters (circle each cluster) all of the angles that are the same in the
drawing, if line X is parallel to line Z.
AD
BC
EH
FG
AE
FB
CG
DH
DE
BG
CF
AH
Use one angle around line X and one angle around line Z. Give all pairs of angles,
whose sum is 180°.
AB
EF
CD
GH
AC
FH
BD
EG
in
Geometry/Trigonometry
Lab
Illustrations
Circles:
1
2 and 3
Polygons:
2
B
C
A
C
3B
D
D
B
A
B
E
3A
A
C
F
E
D
X
Trigonometry:
A
1
2
c
a
θ
b
C
B
D
Z
E
F
G H