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**Pre-calculus lesson 6.1 Law of Sines**

Spring 2011

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**Question! How to measure the depth?**

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**At the end of this lesson you should be able to**

Use the Law of Sines to solve oblique triangles (AAS or ASA). Use the Law of Sines to solve oblique triangles (SSA). Find areas of oblique triangles. Use the Law of Sines to model and solve real-life problems.

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Introduction To solve an oblique triangle, we need to be given at least one side and then any other two parts of the triangle. The sum of the interior angles is 180°. Why can’t we use the Pythagorean Theorem? State the Law of Sines. Area = ½ base*height Area= ½*a*b*sineC A B C a b c Make sure students understand the opposite side and the 2 adjacent sides (vs. adjacent and hypotenuse).

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**Introduction (continued)**

Draw a diagram to represent the information. ( Do not solve this problem.) A triangular plot of land has interior angles A=95° and C=68°. If the side between these angles is 115 yards long, what are the lengths of the other two sides? Make sure students understand the opposite side and the 2 adjacent sides (vs. adjacent and hypotenuse).

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**Playing with a the triangle**

If we think of h as being opposite to both A and B, then A B C a b c h Let’s solve both for h. Let’s drop an altitude and call it h. This means

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A B C a b c If I were to drop an altitude to side a, I could come up with Putting it all together gives us the Law of Sines. Taking reciprocals, we have

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What good is it? The Law of Sines can be used to solve the following types of oblique triangles Triangles with 2 known angles and 1 side (AAS or ASA) Triangles with 2 known sides and 1 angle opposite one of the sides (SSA) With these types of triangles, you will almost always have enough information/data to fill out one of the fractions.

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Example 1 (AAS) A B C a b c I’m given both pieces for sinA/a and part of sinB/b, so we start there. 45° 50° =30 85° Cross multiply and divide to get Once I have 2 angles, I can find the missing angle by subtracting from C=180 – 45 – 50 = 85°

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**We’ll repeat the process to find side c. **

45° 50° =30 85° A B C a b c Make sure students can get the right answer with their calculator. We’ll repeat the process to find side c. Remember to avoid rounded values when computing. We’re done when we know all 3 sides and all 3 angles.

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Example 2 ASA A triangular plot of land has interior angles A=95° and C=68°. If the side between these angles is 115 yards long, what are the lengths of the other two sides? Check for calculator ability

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**The Ambiguous Case (SSA)**

Three possible situations No such triangle exists. Only one such triangle exists. Two distinct triangles can satisfy the conditions.

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Example 3(SSA) Use the Law of Sines to solve the triangle. A = 26, a = 21 inches, b = 5 inches C A b = 5 in a = 21 in 26 B a > b : One Triangle

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Example 3(SSA) Use the Law of Sines to solve the triangle. A = 26, a = 21 inches, b = 5 inches C A b = 5 in a = 21 in 26 B 5.99° a > b : One Triangle

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Example 4(SSA) Use the Law of Sines to solve the triangle. A = 76, a = 18 inches, b = 20 inches A B b = 20 in a = 18 in 76 h a < h:None There is no angle whose sine is There is no triangle satisfying the given conditions.

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**Example 5(SSA) Let A = 40°, b = 10, and a = 9. C**

We have enough information for 40° b = 10 a = 9 h=6.4 A c B h < a < b: Two Cross multiply and divide

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**To get to angle B, you must unlock sin using the inverse.**

40° b = 10 To get to angle B, you must unlock sin using the inverse. 94.4° a = 9 45.6° A B c =14.0 Once you know 2 angles, find the third by subtracting from 180. C = 180 – ( ) = 94.4° We’re ready to look for side c.

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**Example 5 Finding the Second Triangle**

Let A = 40°, b = 10, and a = 9. Start by finding B’ = B Now solve this triangle. b= 10 a = 9 A B’ C’ b= 10 a = 9 40° 134.4° c’ 40° B’ B = 45.6° A B’ = 180 – 45.6 = 134.4°

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**Next, find C’ = 180 – (40 + 134.4) C’ = 5.6° A B’ C’ b= 10 a = 9 40°**

134.4° c Next, find C’ = 180 – ( ) C’ = 5.6° 5.6° =1.4

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**A New Way to Find Area A B C a b c We all know that A = ½ bh.**

And a few slides back we found this. h Area = ½*product of two given sides * sine of the included angle

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**Example 6 Finding the area of the triangle**

Find the area of a triangle with side a = 10, side b = 12, and angle C = 40°.

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Example 7 Application Two fire ranger towers lie on the east-west line and are 5 miles apart. There is a fire with a bearing of N27°E from tower 1 and N32°W from tower 2. How far is the fire from tower 1? The angle at the fire is 180° - (63° + 58°) = 59°. N N x 63° 1 58° 2 5 mi S S Application

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Example 8 Application

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Practice p #s 2-38, even Application

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