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February 2nd, 2013, 04:33 PM  #1 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  find the angles and sides of the triangle
Can you find all other measures of a triangle, if the three sides(SSS) or 2 sides and the included angle (SAS) OR 2 angles and the included sides (ASA) are given? Take for instance a triangle with sides 5cm and 6cm and included angle 60 degrees. Can you find all other angles and sides? 
February 2nd, 2013, 04:47 PM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the angles and sides of the triangle
You should know this! http://www.ehow.com/how_8311180_calcula ... ngths.html 
February 2nd, 2013, 06:29 PM  #3  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: find the angles and sides of the triangle Quote:
 
February 2nd, 2013, 07:06 PM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the angles and sides of the triangle Quote:
Law of Cosines: x^2 = 5^2 + 6^2  2(5)(6)COS(60) Then calculate one missing angle using Law of Sines; then 3rd angle by default.  
February 2nd, 2013, 07:40 PM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond  Re: find the angles and sides of the triangle
Case 1: SSS: Let the triangle have sides a = 4, b = 5 and c = 6 units in length and corresponding angles A, B and C. From the Law of Cosines*, we have 6² = 4² + 5²  2(4)(5)cos(C), where C is the angle between a and b. C = arccos(1/. By the Law of Sines*, sin(C)/c = sin(A)/a, so A = arcsin(asin(C)/c) = arcsin(?63/12), where A is the angle opposite a. A similar method may be used to find B, where B is the angle opposite b. Case 2: SAS: Apply the Law of Cosines to find the length of the side opposite the known angle then use the Law of Sines to compute the remaining angles. Case 3: ASA: Use the fact that the three angles of a triangle sum to ? radians to find the third angle, then use the Law of Sines to compute the remaining sides. *Law of Cosines: The law of cosines states that a² = b² + c²  2bccos(?), where the angle ? is the angle between sides b and c. Let ABC be a triangle with sides a, b and c and corresponding angles A, B and C Let A be the angle opposite a. Then a² = (b  ccos(A))² + (csin(A))² = b²  2bccos(A) + c²cos²(A) + c²sin²(A) = b²  2bccos(A) + c²(cos²(A) + sin²(A)) = b² + c²  2bccos(A). The result is the same for any choice of angle. *Law of Sines: The Law of Sines states that for a triangle ABC with sides a, b and c, the ratios between angles and opposite sides are equal. That is, if A is the angle opposite a, B is the angle opposite b and C is the angle opposite c, then sin(A)/a = sin(B)/b = sin(C)/c. Let ABC be a triangle with sides a, b and c and corresponding angles A, B and C. Construct an altitude h. Then bsin(A) = h = asin(B), so sin(A)/a = sin(B)/b. A similar method may be employed for C and c. 
February 4th, 2013, 04:53 PM  #6 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: find the angles and sides of the triangle
I do it differently; For SSS I don't know any law of cosines, I do it by calculating the area using heron's formula, then find the height and then find angles by calculating tan, sin or any other thing and try finding values of angles that give the closest value of tan or sin For SAS I find the sines or tans or any other thing of the angle and then find the the other side and then the remaining angles. For ASA I find the third angle, then find values of tan, sin.....of the angles and find the sides using them. I think the first one is very different, but the other 2 are some sort of a general procedure of what I do, I think. Any way, thank you very much. The laws of sines and cosines are new to me. Maybe these would become useful to me in the future. 
February 4th, 2013, 05:03 PM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the angles and sides of the triangle
Whatever turns you on, buddy. Law of Sines/Cosines much quicker and simpler. 
February 4th, 2013, 05:13 PM  #8 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: find the angles and sides of the triangle
Can you give a proof of those?

February 4th, 2013, 07:18 PM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the angles and sides of the triangle
Use google. Nobody wants to reinvent the wheel.

February 5th, 2013, 12:36 PM  #10 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond  Re: find the angles and sides of the triangle
. . . or see the bottom of my previous post in this topic . 

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