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Getting StartedThe tasks below are selected with the intent of presenting key ideas and skills. Not every answer iscomplete, so that teachers can still assign these questions and expect students to finish the tasks. If you
are working with your student on homework, please use these solutions with the intention of increasingstudent understanding and independence. A list of questions to use as you work together, prepared in
English and Spanish, is available. Encourage students to refer to their class notes and Math Toolkit entriesfor assistance.
As you read these selected homework tasks and solutions, you will notice that some very sophisticatedcommunication skills are expected. Students develop these over time. This is the standard for which to
strive. See Research on Communication.
The Geometry page or the Scope and Sequence (2nd edition) might help you follow the conceptual
development of the ideas you see in these examples.
Main Mathematical Goals for Unit 7Upon completion of this unit, students should be able to:
• explore the sine, cosine, and tangent functions defined in terms of a point on the terminal side of
an angle in standard position in a coordinate plane.
• explore properties and applications of the sine, cosine, and tangent ratios of acute angles in right
triangles.
• determine measures of sides and angles for nonright triangles using the Law of Sines and Law of
Cosines.
• use the Law of Sines and Law of Cosines to solve a variety of applied problems that involve
triangulation.
• describe the conditions under which two, one, or no triangles are determined given the lengths of
two sides and the measure of an angle not included between the two sides.
What Solutions are Available?Lesson 1: Investigation 1—Applications Task 1 (p. 474), Connections Task 9 (p. 477),
(These solutions are for tasks in the 2nd edition book—2008 copyright.For homework tasks in books with earlier copyright dates, see Helping with Homework.)
c. Since the trigonometric values result from ratios of the coordinates and the radius of the circle, you
can find the trigonometric values for angles in Quadrant II (obtuse angles) by using the image of thepoint on the circle reflected across the y-axis. So, for example, a point (–x, y) in Quadrant II has
image point (x, y) in Quadrant I. The negative x-coordinate in Quadrant II means that the cosine andtangent values will be negative in Quadrant II. So, to use the table on page 465 for 120˚ or P12, reflect
P12 across the y-axis to P6. Use the negative of the cos 60˚ and tan 60˚ function values from the tablefor a 120˚ angle. The sin 60˚ = sin 120˚.
a. Students should verify that parallelograms have 180˚ rotational symmetry. (The inclusion of thediagonals in their analysis of the symmetry will assist them in finding some of the information for
If your student has done Connections Task 9, the area formula for a triangle given in that task can be usedhere. The solution given below does not use that formula. This solution uses the standard area formula for
a triangle, area = 0.5(base height).
c. In ABD, the length of the altitude from point A can be found by solving sin 19˚ = h27 . So,
h = 27 sin 19˚, and the area of ABD = 12 (43.5)(27) sin 19˚ 191 m2. In CBD, the length of the
altitude from point B can be found by solving sin 53˚ = h24 . So, h = 24 sin 53˚, and the area of
CBD = 12 (53.5)(24) sin 53˚ 513 m2. Area ABCD 191 m2 + 513 m2 = 704 m2.
(Old Glory’s record throw is listed as 4,224.00 feet on the Punkin’ Chunkin’ Web site:
www.atbeach.com/punkinchunkin/)
INSTRUCTIONAL NOTE If students use the Law of Sines in Part b to determine the measure of J,
they will get the measure of the supplement of J. You can refer students back to Lesson 1, ConnectionsTask 8, to help them make sense of this situation. From Connections Task 8, they know that two possible
angles between 0˚ and 180˚ exist for any given sine value (except for sin 90˚ = 1). Therefore, using sin–1
to determine the measure of an angle in any triangle should always reveal two possible solutions. To
avoid this dilemma, students can use the Law of Cosines.