English 9 - PLATO Learning Inc.



Applying the Laws of Sines and Cosines

The Lesson Activities will help you meet these educational goals:

• Content Knowledge—You will understand and apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.

• STEM—You will apply mathematical and technology tools to analyze real-world situations.

• 21st Century Skills—You will carry out technology-assisted modeling.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities[pic]

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1. Solving Non-Right Triangles

You will use an online tool to solve non-right triangles. Go to the triangle solver, and complete each step below.

a. Given a non-right triangle with limited information, you will apply the Law of Sines and the Law of Cosines to find the remaining side lengths and angle measures. Click the New Problem button to find a problem in which you are given a non-right triangle with one known side length and two known angles. Find the length of one of the unknown sides using the Law of Sines. Take a screenshot of your result, and paste it below.

Type your response here:

b. Click the New Problem button to generate a problem with a non-right triangle in which two side lengths and an angle opposite to one of the two sides are given. Solve for one of the unknown angles using the Law of Sines. Take a screenshot of your result, and paste it below.

Type your response here:

c. Click the New Problem button again to find a problem with a non-right triangle in which two side lengths and the included angle are given. Find the unknown side length using the Law of Cosines. Paste a screenshot of your result below.

Type your response here:

d. Finally, click the New Problem button again to find a problem with a non-right triangle in which all three side lengths of the triangle are given. Find one of the angles using the Law of Cosines. Paste a screenshot of your result below.

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

2. Applying the Laws of Sines and Cosines

You will use the GeoGebra geometry tool to construct a triangle with three locations whose bearings relative to each other are known and among which one pair of distances is known. Go to the coordinate map for Panama, and complete each step below. If you need help, follow these instructions for using GeoGebra.

You know that Cali, Columbia, is 435 miles at a bearing of S 20° E from Panama City, Panama. The bearing of San Cristobal, Venezuela, is S 81° E from Panama City, and the bearing of Cali from San Cristobal is S 53° W. The vertical line in the diagram is the north-south line that passes through point P (Panama City).

a. Follow these steps to construct a triangle to represent the given information:

• Draw a ray in the direction S 20° E from point P. Create a circle with radius 43.5. (Note that you’ll be scaling down all the distances in miles by a factor of 10: 435 ÷ 10 = 43.5.) Label the point where the circle intersects the ray as C (Cali).

• Draw a ray in the direction S 81° E from P. Mark an arbitrary point on this ray, and label it S.

• Create a line through point S that is parallel to the north-south line through P.

• Using the line passing through point S, draw a ray that is S 53° W.

• Move point S along [pic] until the ray S 53° W from S passes through point C. This position of S locates San Cristobal.

Paste a screenshot of your construction in the space below.

Type your response here:

b. Use GeoGebra to find m[pic] and m[pic] in [pic].

Type your response here:

c. Based on the known information about [pic] what relationship can you use to find the length of [pic] which represents the distance between San Cristobal and Panama?

Type your response here:

d. Find the length of [pic] the distance between San Cristobal and Panama. Show your work. Remember to scale the distance up by a factor of 10.

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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Lesson Activities

Geometry

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