Edmentum Integrated Math3 Unit 5 Activity: Trigonometry and Geometric Modeling

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You can often use geometric figures to model objects in the real world. You can transfer your knowledge of the properties of these figures to better understand and describe the objects that they represent.

For each shape in the table, list three examples of real-world objects that could be modeled by the shape. Use your experiences, the Internet, newspapers, magazines, or other resources to uncover examples.

Gordon works for a graphic design firm and is creating a label for a food truck vendor. The vendor specializes in finger food and wants to sell food in right conical containers so that they are easy for people to hold. To complete his label, Gordon needs to collect several different measurements to ensure that the label he designs will fit the surface of the container. Gordon has been told that the containers have a diameter of 4 inches and a height of 6 inches.

Find the slant height of the cone. The slant height is the distance from the apex, or tip, to the base along the cone’s lateral surface. Show your work.

Find the measure of the angle formed between the base of the cone and a line segment that represents the slant height.

Imagine two line segments where each represents a slant height of the cone. The segments are on opposite sides of the cone and meet at the apex. Find the measurement of the angle formed between the line segments.

A vector is a quantity that has magnitude and direction. For example, if you travel 20 miles northwest, 20 miles is the magnitude and northwest is the direction. In this example, the vector is called a displacement vector. Vectors often represent displacement, speed, acceleration, or force.

You can think about a vector as a directed line segment. The initial point is the tail of the vector. The terminal point is the tip, usually represented by an arrowhead. The vector in the diagram can be named either or .

You can also describe a vector using component form. This form defines the vector according to the horizontal and vertical changes in the coordinates from the initial point to the terminal point. If x represents the horizontal change of and y represents the vertical change of then the component form of is x, y. In the figure above, the horizontal change of is and the vertical change is . Therefore, in component form, .

You will now use this basic understanding of vectors to answer some questions about the magnitude of a vector. You may use the GeoGebra geometry tool to assist you with your answers, but using the program is not required. If you need help, follow these instructions for using GeoGebra

How can you find the magnitude of a vector , where the horizontal change is x and the vertical change is y?

What is the magnitude of the vector

The direction of a vector is defined as the angle of the vector in relation to a horizontal line. As a standard, this angle is measured counterclockwise from the positive x-axis. The direction or angle of in the diagram is α.

How can you use trigonometric ratios to calculate the direction α of a general vector , similar to the diagram?

Suppose that vector lies in quadrant II, quadrant III, or quadrant IV. How can you use trigonometric ratios to calculate the direction (i.e., angle) of the vector in each of these quadrants with respect to the positive x-axis? The angle between the vector and the positive x-axis will be greater than in each case.

Now try a numerical problem. What is the direction of the vector ?

Two vectors are said to be parallel if they point in the same direction or if they point in opposite directions.

Are the vectors and parallel? Show your work and explain.