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Math 20: Foundations FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case). D. The Trigonometric Legal Department

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Starting Point Lacrosse Trigonometry p.114

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1. How are Sides and Angles of a Triangle Related? FM20.5 Demonstrate understanding of the cosine law and sine law.

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What are two equivalent expressions that represent the height of ∆ABC ?

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Summary p.117

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Practice Ex. 3.1 (p.117) # 1-5

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2. The Law of Sines FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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2. The Law of Sines Last day we discovered a side-angle relationship in acute triangles. Before we can use this to solve problems we have to prove it is true for all acute triangles

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Example 1

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Example 2

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**Note:

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Example 3

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Summary p.124

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Practice Ex. 3.2 (p.124) #1-15 #4-19

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3. The Cosine Law! FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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3. The Cosine Law! Unfortunately the Sine Law will not work for all situations.

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For example.

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For these situations another Relationship was created called the Cosine Law Again we must prove it before we can use it.

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The Cosine Law is: Used when you have: 2 sides and included angle All 3 sides.

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Explain why you can use the cosine law to solve for side q in ∆QRS and for ∠F in ∆DEF on page 130.

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Example 1

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Example 2

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Example 3

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Summary p.137

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Practice Ex. 3.3 (p.136) #1-14 #4-17

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4. Use Triangles to Solve the Problem FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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4. Use Triangles to Solve the Problem

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Example 1

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Example 2

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Summary p.146

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Practice Ex. 3.4 (p.147) #1-14 #3-17

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5. What about Obtuse Triangles FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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* Until now, you have used the primary trigonometric ratios only with acute angles. For example, you have used these ratios to determine the side lengths and angle measures in right triangles, and you have used the sine and cosine laws to determine the side lengths and angle measures in acute oblique triangles.

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Let’s investigate the values of the primary trig ratios for obtuse triangles Evaluate sin100° However there is not right triangle that can be made with a 100° angle

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However we can make a right triangle with its supplement 80° What is sin80°?

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Complete the chart on p. 163 and the reflection questions that follow.

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Summary p.163

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Practice Ex. 4.1 (p.163) #1-4

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6. Do the Sin and Cos Laws Work for Obtuse Triangles? FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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6. Do the Sin and Cos Laws Work for Obtuse Triangles? In section 3 we proved the Sin Ratio for Acute triangles. We are going to adjust that proof to prove the Sin Law for Obtuse Triangles.

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Example 1

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Example 2 Determine the distance between Jaun and the Balloon.

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Let’s now look to see if the Cosine Law holds true for Obtuse Triangles

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Example 3

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Summary p.170

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Practice Ex. 4.2 (p.170) #1-5 evens in each, 6-15 #3-17

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7. How Many Triangles Exist? (The Ambiguous Case) FM20.5 Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

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7. How Many Triangles Exist? (The Ambiguous Case) Ambiguous Case – A situation in which two triangles can be drawn, given the available information; the ambiguous case may occur when the given measurements are the lengths of two sides and the measure of an angle that is not contained by the two sides (ASS).

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So when you are dealing with the Ambiguous Case and given SSA as your information there are 4 possible solutions 1 Oblique Triangle 1 Right Triangle 2 Triangles No Triangles

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How do we figure out how many triangles are possible given the Ambiguous Case (SSA)?

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Example 1

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Example 2

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Example 3

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Practice Ex. 4.3 (p.183) #1-14 #4-17

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8. Many Problems Involve Obtuse Triangles

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Example 1

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Summary p.193

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Practice Ex. 4.4 (p.193) #1-14 #2-16

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