## Course 2, Unit 6 - Geometric Form and Its Function

Summary
Need summary here, like that provided for Course 2 Units 1-4.

Key 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 scale images.

• 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 similar.
• 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 example, if 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.
1. r1 = r2: When the radii are equal, the angular velocity of the second circle is equal to that of the first.
2. 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.
3. 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 a factor 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 360/b (if 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.
Copyright 2013 Core-Plus Mathematics Project. All rights reserved.