2, Unit 6 -
Geometric Form and Its Function
Need summary here, like that provided for Course 2 Units 1-4.
Ideas from Course 2, Unit 6
- Parallelogram linkage: A parallelogram linkage has two pairs
of parallel sides, connected in a way that fixes one side, called the frame.
The side opposite the frame is called a coupler, and the other
two sides are called cranks. All connections are flexible, allowing
the angles of the parallelogram to change. These linkages are used
to drive useful objects like wheels, which make complete rotations,
or windshield wipers, which turn through a prescribed angle, or create
- Two plane shapes are similar with a scale factor k: if
the lengths of the segments of the second shape are k times
the lengths of the segments of the first plane shape. Measures of corresponding
angles are equal. The area of the second shape is k2 times
the area of the original shape. For example:
Lengths of sides of the larger parallelogram are twice the lengths of
sides of the smaller parallelogram. The area of the larger parallelogram
is four times larger than the smaller. Corresponding angles are equal.
The parallelograms are
- The sine, cosine, and tangent of the acute angle of a right triangle: sin C = (length
of leg opposite angle C)/(length of hypotenuse); cos C = (length
of leg adjacent to angle C)/(length of hypotenuse); tan C = (length
of leg opposite angle C)/(length of leg adjacent to angle
C). If you know the lengths of two sides, or the measure of an
angle and length of a side, of a right triangle, you can find the
lengths of all other sides. For
sin C = c/b,
then if we know lengths c and b, we can find sin C and
then take an inverse sine to find the measure of angle C.
If sin C = c/b and
we know the measure of angle C and length b, then c = b sin C,
so we can find length c. For example:
If we know that angle C is 43 degrees, then we can set up tan 43 = opp/adj = ?/11, and find the missing side. AB = 11(tan 43) = 10.26 (approx).
- Angular velocity: The rate of change of the angle as a radius
sweeps out a circle. This is used in examples like pulleys where two
wheels are connected. The angular velocities of the two wheels depend
on whether the radii of the wheels are equal or not.
- r1 = r2: When the radii are
equal, the angular velocity of the second circle is equal to that of
- r1 > r2: When the first radius
is larger, then the angular velocity of the second circle is increased by
a factor of r1/r2. This is the situation
with a bike gear; when the gear wheel being directly turned by the user
is 2 times larger than the connected gear, this makes the second gear turn
2 times more frequently and, therefore, increases efficiency.
- r1 < r2: When the first radius
is smaller, then the angular velocity of the second circle is decreased
by a factor of r1/r2. In the situation
with a bike gear, if the smallest gear wheel being turned is a third the
size of the connected gear, this means the second (larger) gear turns at
one third the frequency.
- One radian: The amount of rotation or angle size that results
when a radius of any size sweeps out an arc equal to the radius.
Since we need 2π radii to wrap around a circle, we know that
2π radians = 360 degrees.
One radian is approximately 60 degrees.
- Period of a trigonometric function: is 2π in a standard
function like y = sin x. That is, as x changes
from 0 to 2π, the y values change through all possible
values of y.
As x changes from 2π to 4π, the y values
will repeat. To have a period different from 2π, we have to introduce
into the equation. In general, the period of the function y = a sin bx is
2π/b (if the unit of measure is radians), or
the unit of measure is degrees).
- Amplitude of the equation: The y values of a sine or cosine function vary between a maximum and minimum. The amplitude is half of the difference between the maximum and minimum values. For a standard function y = sin x, the y values vary from -1 to 1, so the amplitude is 1. In the function y = a sin x, the amplitude is a.