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Law of Sines Ambiguous Case Grade: 11 Subject: Pre-‐Calculus 11 Unit: Trigonometry Learning Target: Develop an understanding of the ambiguous case. Explore three different cases and determine how they are related. Curriculum Outcomes:
GCO’s SCO’s Students will be expected to develop trigonometric reasoning.
AN01: Students will be expected to solve problems, using the cosine law, and the sine law, including the ambiguous case.
Expected Time: 1-‐2 classes Resources: (Tools & Tech)
Lesson Procedure
We Do: Law of Sines Ambiguous Case Activity. This activity should be completed before students receive an introduction of the ambiguous case. Students should discover on their own that with some sides lengths, two possible triangles can be created. Some students will need some prompts to see this as they will fall into the trap of assuming a pattern and not examining other options. Teacher directions and suggestions have been provided with further explanation than is provided on the student directions sheet. The idea is that students will discover the ambiguous case on their own and be able to visualize the different scenarios once
the theory is presented.
I Do / You Do: Student Worksheet is provided. The chart should be filled out as a class discussion, and an example is provided on the sheet for applying the concept of the ambiguous case to solving triangles. Solutions to student worksheet have been provided.
You Do: Practice Problems to reinforce the concept of ambiguous case.
LAW OF SINES: AMBIGUOUS CASE ACTIVITY
Materials Needed: Strips of paper or pipe cleaners (2 different colours) Protractor Ruler 11 x 17 piece of paper Directions:
• Draw a broken line near the bottom of the paper; this will be the base line of the triangle.
• Cut one strip of paper 16 cm long. Attach it to the baseline at a 30° angle to
form one arm of the triangle. This strip will not be adjusted so it should be taped into position.
• Start with a second strip that is 24 cm long.
• As triangles are created, draw a sketch of the triangle on the paper.
• Shorten the strip by 1 cm increments, drawing the all the triangles that can be created as you go.
Hint: There will be more than 20 possible triangles!!
LAW OF SINES: AMBIGUOUS CASE ACTIVITY
Teacher Directions
Materials Needed: Strips of paper or pipe cleaners (2 different colours) Protractor Ruler 11 x 17 piece of paper Directions:
• Draw a broken line near the bottom of the paper; this will be the base line of the triangle.
• Cut one strip of paper 16 cm long. Attach it to the baseline at a 30° angle to
form one arm of the triangle. This strip will not be adjusted so it should be taped into position.
• Start with a second strip that is 24 cm long.
• As triangles are created, draw a sketch of the triangle on the paper.
• Shorten the strip by 1 cm increments, drawing the all the triangles that can be created as you go.
Notes: Once the strip becomes shorter than the fixed arm of the triangle, there will be two possible triangles for each arm length. When the strip is half the length of the fixed arm, the triangle formed will be right angled. (𝑎 = 𝑏𝑠𝑖𝑛𝐴) When the strip is shorter than half the length of the arm (𝑎 < 𝑏𝑠𝑖𝑛𝐴), no possible triangle can be formed. Have students identify the total number of triangles that were created and for which lengths, there were two possible triangles.
LAW OF SINES: AMBIGUOUS CASE
Sketch of Possible Triangle(s) Number of Solutions
a > b
a = b
a < b
Ex. Solve ∆ ABC if ∠ A = 27° , a = 3.2 cm , b = 4.1 cm.
Practice Problems: For each of the following questions,
• Indicate whether the given measurements result in no triangle, one triangle, or two triangles.
• Draw a sketch of the possible triangle(s). • Solve the resulting triangle. Round the answer to the nearest tenth.
1. ∠ A = 22°, a = 16.8 cm, b = 22.42 cm 2. ∠B = 96°, b = 3 cm, a = 24 cm
3. a = 9 cm, b = 7 cm , ∠A = 49°
Solutions: Sketch of Possible Triangle(s) Number of Solutions
a > b
1 possible triangle
a = b
1 possible triangle
a < b
2 solutions if 𝒂 > 𝒃𝒔𝒊𝒏𝑨 1 solution if 𝒂 = 𝒃𝒔𝒊𝒏𝑨 No Solution if 𝒂 < 𝒃𝒔𝒊𝒏𝑨
Ex. Solve ∆ ABC if ∠ A = 27° , a = 3.2 cm , b = 4.1 cm. a < b: Ambiguous Case a = 3.2 , bsinA = 1.86 a > bsinA: 2 solutions
∠A = 27°
a = 3.2 cm
∠B = 36°
b = 4.1 cm
∠C = 117°
c = 6.3 cm
∠A = 27°
a = 3.2 cm
∠B = 144°
b = 4.1 cm
∠C = 9°
c = 1.1 cm
Practice Problems:
1) Two possible triangles:
Solution 1: ∠B = 30°, ∠C = 128°, c = 35.3
Solution2: ∠B = 150°, ∠C=8°, c = 6.2
2) No triangle exists.
3) One triangle
∠B = 35.94°, ∠C = 95.06°, c = 11.88