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Solving Triangles I
Ó 2006, Frank Ma
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Solving Triangles I
Objective:
*  Use Sin= e Law and Cosine Law to solve  = ;
    triangles for the determined cases:
    ASA, SAS, SSS
*  Identif= y and solve the ambiguous
   ca= ses: AAA, ASS
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Some Basic Geometric Facts of Triangles
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Some Basic Geometric Facts of Triangles
Standard labeling of a triangle
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Some Basic Geometric Facts of Triangles
<= span style=3D'font-size:88%'>•  A triangle is rigid.
Standard labeling of a triangle
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Some Basic Geometric Facts of Triangles
•<= /span>  A t= riangle is rigid.
 
= • (Triangle Inequality) The <= /span><= span style=3D'font-size:88%'>sum of the lengths of any two <= span style=3D'font-size:88%'>sides is more than the length <= span style=3D'font-size:88%'>of third side.
Standard labeling of a triangle
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Some Basic Geometric Facts of Triangles
•<= /span>  A t= riangle is rigid.
 
= • (Triangle Inequality) The <= /span><= span style=3D'font-size:88%'>sum of the lengths of any two <= span style=3D'font-size:88%'>sides is more than the length <= span style=3D'font-size:88%'>of third side.
• The longest side is     <= span style=3D'position:absolute;top:55.75%;left:4.3%;width:60.29%;height:6.0%'>= opposite to the largest angle. <= /div>
<= span style=3D'font-size:88%'>The medium side is opposite <= span style=3D'font-size:88%'>to the medium angle.
= The shortest side is opposite <= span style=3D'font-size:88%'>to the smallest angle.
Standard labeling of a triangle
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0110.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Some Basic Geometric Facts of Triangles
•<= /span>  A t= riangle is rigid.
 
= • (Triangle Inequality) The <= /span><= span style=3D'font-size:88%'>sum of the lengths of any two <= span style=3D'font-size:88%'>sides is more than the length <= span style=3D'font-size:88%'>of third side.
• The longest side is     <= span style=3D'position:absolute;top:55.75%;left:4.3%;width:60.29%;height:6.0%'>= opposite to the largest angle. <= /div>
<= span style=3D'font-size:88%'>The medium side is opposite <= span style=3D'font-size:88%'>to the medium angle.
= The shortest side is opposite <= span style=3D'font-size:88%'>to the smallest angle.
•<= /span> Sum = of all angles is 180o
Standard labeling of a triangle
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Solving Triangles
Given information about sides and angles of a triangle,= = we may try to solve= for the missing angles or sides.
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Solving Triangles
Given information about sides and angles of= a triangle, we may try to solve for the missing angles or sides.
We called this solving for the triangle<= span style=3D'font-size:156%;color:black;mso-color-index:3'>.
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Solving Triangles
Given information about sides and angles of= a triangle, we may try to solve for the missing angles or sides.
We called this solving for the triangle<= span style=3D'font-size:156%;color:black;mso-color-index:3'>.
They can be grouped= in the following cases.
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Solving Triangles
I.  Determined Cases
ASA: Angle-Side-Angle
SAS: Side-Angle-Side
SSS: Side-Side-Side
Given information about sides and angles of= a triangle, we may try to solve for the missing angles or sides.
We called this solving for the triangle<= span style=3D'font-size:156%;color:black;mso-color-index:3'>.
They can be grouped= in the following cases.
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Solving Triangles
I.  Determined Cases
ASA: Angle-Side-Angle
SAS: Side-Angle-Side
SSS: Side-Side-Side
II. Ambiguous Cases= :
AAA: Angle-Angle-Angle
        (Similar triangles)
ASS: Angle-Side-Side
        (Ambiguous Cases)
Given information about sides and angles of= a triangle, we may try to solve for the missing angles or sides.
We called this solving for the triangle<= span style=3D'font-size:156%;color:black;mso-color-index:3'>.
They can be grouped= in the following cases.
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ASA: Angle-Side-Angle = ;
Solving Triangles
       Determined Cases
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Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. = ;
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0229.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0118.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Solving Triangles
       Determined Cases
c
draw c=
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
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c
draw c=
draw &= lt;A
A
Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0119_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhpACWAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCg AJMAgAAAAAAAAAL/hI+py+0PQYi02ouz3ilMDobiSDbeV6bqyiIn2sbyTL0wjef5qfc+ffsJh8Si 8YhMFoPK5tIJRTKj1Kr1is1qt1zRtAvGfMNkyLiMZpzT7Lb7DY/LOes5uG7f4vP8vv8PGJiyJ+hE WNh0iJikuOj4CBkpqdQ42VNpqYOZibPJ+QkaOuQpukJaOojKqMra6voK6AE7Oktbe4ubq7u629kL 9AscPExcbPxweqyQrHzA3AwdLe333FxtPR1ync3drem9sW0sPg5ufo7uRl68Ptzunh4vP2/1Hmwf h++in2ZD3+Hvn4GAAm3wc3CwHsEjCaMspPTIYD1EEsM17PPQIg9dqRUHZdTGp2MMkSAuzvgIBGUF WepUXnKJsA1JRv5YYnyhxaDJkzCr9OQys0vQgTlxqquh0OiinQh/enPabWg6qZFMQo34UqkqfVRD vbuaras5seDAsiJF9hamtLgamW278m00ttzkwgOolQpTYQPt/rA5Z4LfURvlDF5S2DCbw9ASC1y2 txXgx/soL5hsmWhmgJsrd+77GXRoCaMlRD57emtqVJgzr2a9ugAAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0120.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
c
draw c=
draw &= lt;A
A
draw = <B
B
Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0120_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhpACWAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCg AJMAgAAAAAAAAAL/hI+py+0PQYi02ouz3ilMDobiSDbeV6bqyiIn2sbyTL0wjef5qfc+ffsJh8Si 8YhMFoPK5tIJRTKj1Kr1is1qt1zRtAvGfMNkyLiMZpzT7Lb7DY/LOes5uG7f4vP8vv8PGJiyJ+hE WNh0iJikuOj4CBkpqdQ42VNpqYOZibPJ+QkaOuQpukJaOojKqMra6voK6AE7Oktbe4ubq7u629kL 9AscPExcbPxweqyQrHzA3AwdLe333FxtPR1ync3drem9sW0sPg5ufo7uRl68Ptzunh4vP2/1Hmwf h++in2ZD3+Hvn4GAAm3wc3CwHsEjCaMspPTIYD1EEsM17PPQIg9dqRUHZdTGp2MMkSAuzvgIBGUF WepUXnKJsA1JRv5YYnyhxaDJkzCr9OQys0vQgTlxqquh0OiinQh/enPabWg6qZFMQo34UqkqfVRD vbuaras5seDAsiJF9hamtLgamW278m00ttzkwgOolQpTYQPt/rA5Z4LfURvlDF5S2DCbw9ASC1y2 txXgx/soL5hsmWhmgJsrd+77GXRoCaMlRD57emtqVJgzr2a9ugAAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0120_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhpABMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwCf AEcAgAAAAAAAAALmjA2py+0Po5y02ovBQbn7D4aiswXjiaaqx63uC8daK9f2HZn4zvf+DzwdgsQi RmdMKnPLppPmjAaR0urUiu1Bs9yu99vYgsefIfmMTgPF6jaE7Y7L5xY43U296zX7u7k/lwc4SEj2 V5gmiJi42GbXCBn58yhZRVmJmblyqZl02AkaOsIpOqVYipoqQarKc9rqCdv06cOK+Xpjq4kbo7vr CwIs6yIcyit0PDwjU6yM0pwKzVDifCGdsFFtmqwd1nw9/N097shNTmx+Tqxem84u9D4Zj0M7bx/n fs8Crv+Q39+BH0BvBQAAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0121.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
c
draw c=
draw &= lt;A
A
draw = <B
B
Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
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c
draw c=
draw &= lt;A
A
draw = <B
B
To solve for the triangle, we use the <= span style=3D'font-size:156%;color:red'>sine law. = ;
Solving Triangles
       Determined Cases
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
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Sine Law:  For any triangle,
sin(A)
sin(B)
sin(C)
=3D
=3D
a
b
c
or
sin(A)
sin(B)
sin(C)
=3D
=3D
a
b
c
ASA: Angle-Side-Ang= le
Given two angles an= d the side between them, it determines a triangle. If angles A, B and side c are given, build the tr= iangle as shown:
c
draw c=
draw &= lt;A
A
draw = <B
B
To solve for the triangle, we use the <= span style=3D'font-size:156%;color:red'>sine law. = ;
Solving Triangles
       Determined Cases
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
16
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0133_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC7 AAIAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7XQAAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
16
29
A
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0130_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC7 ADYAgAAAAAAAAALShI+py+0Po5y00hCs3rz7Dz5YSJbmiTZjyrbuC63wTNdgZuf6Hsn8D6zhgsRi CzM0KpecJPMJlSCj1KrCac1SfdquEusNB6fisvmM9iDB6bb7XSbD5yQ2/b6R4/f8Pm3tF9gjSBjD VYiYqJgAuOj4WKgHKWg3SXdoyYeZOSfJuVf52bYpCkdainaKmqa6GueK5wl71jobVmubFZrrJcvb +/uGGwy1S1zle4ysTDvM/AxtkBy9ZExttJatvc3d7f0NHi4+Tl5ufo6err7O3u7+jlEAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0131.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
16
29
74
C
A
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
16
29
74
A
C
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
16
29
74
A
C
c
a
B
To solve the triangle:
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
<B =3D 180 = 211; 29 – 74 =3D 77
16
29
74
A
c
a
B
To solve the triangle:
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
16
29
74
A
C
c
a
B
To solve the triangle:
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
so a =3D
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
sin(77)
16sin(29)<= span lang=3DEN-US style=3D'font-size:156%;color:#006600;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
16
29
74
A
C
c
a
B
To solve the triangle:
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
so a =3D
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
sin(77)
16sin(29)<= span lang=3DEN-US style=3D'font-size:156%;color:#006600;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
» 7.96
16
29
74
A
C
c
a
B
To solve the triangle:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACP AAEAgAAAAAAAAAIOjI+py+0Hopy01oezZgUAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAg ABwAgAAAAAAAAAJAjI+pm+APo5zO0IufzVzuDgJfyI0kZp5UqnpBm7Gw9s5rbbv5JNv9/IMFW0NV 8XQkJUNLULPzLOF2NGqEgc0qCgA7 ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy ACAAgAAAAAAAAAJYRI6py+0Kopy02ovlybz7uH3iCAbk6YXoaqnsW8KyYc6va594Pu785/t1gsIM sXg5IivK5aTpjEWHtamxak1is8wt9+n9SsUUaNTsRC/VSHbRLXzI53RFAQA7 ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgBXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC7 AFYAgQAAAAAAAP8AAAECAwL/hI+py+0Po5wo0Iuz3rz7ZX3iSJbmFp7qyrZZ6sbyTNf2jef6zktw DwyqfsKisUM8KpfMpvMJjUqnVEyyis1ds9zZtgsOi8fksutrTr/UbBG6DY/L5/S6/fi+y/N6OL8P GCg4aPZHOGZ4CJao2MXYeBfwCBklSTn1ZnmZtaW5yQUz+dnkObpoSiYpihr0tcpao/rwCtvCSFtr UpqLyQu66zuIG8ygOtxwTKyMJbv8lOx8BqwDHR1THWhMGo1tfdLsHa6sLdYt/t0Izmbee47H7o4E 7zTPoy7cVk+l78UffzHtnxZ/An0QLPTs3MEJ97wtRPawIESJQCLio9GQYoJ5QRYvsQuo0QG0jiEN ZCwpg+Q4gCBRclDpEkCnljGH1MShSoDOnTx7+vwJNKjQoUSLGj2KNKnSpUybOn0KNapUpgUAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0137_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 ADMAgAAAAP8AAAL/hI+py+0Po5xUhoqz3rx7HYTiR5bmiVriBYTpC8fys7KIO+f6vtUOzgsKh74I cIhMnoqTo/IJpTAxzqj1apj2bNgussb9VL1kGPg1LqvF2lR6DZe2Y++4fTHP1e/3PG/PV+ZHFBao NqgEaJiECKW4+LfC9whJJ7lIWVlyptlS2InCCeo56taI+VmacdqZqZrAOur6Gkub+gp7iSu3i6fb WzHb9wtMdVtZW9wg3CWqzIac/KwS6Dy9dHxlfW16SMwtw2z2DT4jjj1S/pUdTq4edA7i/k6oI03f 3p6O7xXvO8/Pir8s9wLW2wLQYD92/wYqPGgk4cM4ewpObJbK4kUyDkc0bvTmyeFHQSJHwikAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0138.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
so a =3D
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
sin(77)
16sin(29)<= span lang=3DEN-US style=3D'font-size:156%;color:#006600;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
» 7.96
sin(74)
sin(77)
c
16
=3D
16
29
74
A
C
c
a
B
To solve the triangle:
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•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
so a =3D
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
sin(77)
16sin(29)<= span lang=3DEN-US style=3D'font-size:156%;color:#006600;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
» 7.96
sin(74)
sin(77)
c
16
=3D
so c =3D
sin(77)
16sin(74)
16
29
74
A
C
c
a
B
To solve the triangle:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACP AAEAgAAAAAAAAAIOjI+py+0Hopy01oezZgUAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAg ABwAgAAAAAAAAAJAjI+pm+APo5zO0IufzVzuDgJfyI0kZp5UqnpBm7Gw9s5rbbv5JNv9/IMFW0NV 8XQkJUNLULPzLOF2NGqEgc0qCgA7 ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACP AAEAgAAAAAAAAAIOjI+py+0Hopy01oezZgUAOw== ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy ACAAgAAAAAAAAAJYRI6py+0Kopy02ovlybz7uH3iCAbk6YXoaqnsW8KyYc6va594Pu785/t1gsIM sXg5IivK5aTpjEWHtamxak1is8wt9+n9SsUUaNTsRC/VSHbRLXzI53RFAQA7 ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQBZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgBXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC7 AFYAgQAAAAAAAP8AAAECAwL/hI+py+0Po5wo0Iuz3rz7ZX3iSJbmFp7qyrZZ6sbyTNf2jef6zktw DwyqfsKisUM8KpfMpvMJjUqnVEyyis1ds9zZtgsOi8fksutrTr/UbBG6DY/L5/S6/fi+y/N6OL8P GCg4aPZHOGZ4CJao2MXYeBfwCBklSTn1ZnmZtaW5yQUz+dnkObpoSiYpihr0tcpao/rwCtvCSFtr UpqLyQu66zuIG8ygOtxwTKyMJbv8lOx8BqwDHR1THWhMGo1tfdLsHa6sLdYt/t0Izmbee47H7o4E 7zTPoy7cVk+l78UffzHtnxZ/An0QLPTs3MEJ97wtRPawIESJQCLio9GQYoJ5QRYvsQuo0QG0jiEN ZCwpg+Q4gCBRclDpEkCnljGH1MShSoDOnTx7+vwJNKjQoUSLGj2KNKnSpUybOn0KNapUpgUAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0139_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 ADMAgAAAAP8AAAL/hI+py+0Po5xUhoqz3rx7HYTiR5bmiVriBYTpC8fys7KIO+f6vtUOzgsKh74I cIhMnoqTo/IJpTAxzqj1apj2bNgussb9VL1kGPg1LqvF2lR6DZe2Y++4fTHP1e/3PG/PV+ZHFBao NqgEaJiECKW4+LfC9whJJ7lIWVlyptlS2InCCeo56taI+VmacdqZqZrAOur6Gkub+gp7iSu3i6fb WzHb9wtMdVtZW9wg3CWqzIac/KwS6Dy9dHxlfW16SMwtw2z2DT4jjj1S/pUdTq4edA7i/k6oI03f 3p6O7xXvO8/Pir8s9wLW2wLQYD92/wYqPGgk4cM4ewpObJbK4kUyDkc0bvTmyeFHQSJHwikAADs= ------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0140.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
•Example A:  Solve for
•the triangle given that
•<A=3D29o, b=3D16, <C=3D74o.
•Draw and label.&n= bsp;
ASA and the Sine law
sin(29)
sin(77)
so a =3D
a
16
<B =3D 180 = 211; 29 – 74 =3D 77
=3D
sin(77)
16sin(29)<= span lang=3DEN-US style=3D'font-size:156%;color:#006600;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
» 7.96
sin(74)
sin(77)
c
16
=3D
so c =3D
sin(77)
16sin(74)
» 15.8
16
29
74
A
C
c
a
B
To solve the triangle:
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•Example:  Solve the
•triangle given that
•<B=3D31o, c=3D16, <A=3D114o.
•Draw and label.&n= bsp;
Your Turn: ASA and the Sine la= w
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•Example:  Solve the
•triangle given that
•<B=3D31o, c=3D16, <A=3D114o.
•Draw and label.&n= bsp;
Your Turn: ASA and the Sine la= w
A
B
c=3D16
114
31
35
a
b
a =3D 16*sin(114)/sin(35) » 25.5
b =3D 16*sin(31)/sin(35) »14.4
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SAS: Side-Angle-Side
Solving Triangles
       Determined Cases
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SAS: Side-Angle-Sid= e
Given t= wo sides and the angles between them, it determines a triang= le.
Solving Triangles
       Determined Cases
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Solving Triangles
       Determined Cases
SAS: Side-Angle-Sid= e
Given t= wo sides and the angles between them, it determines a triang= le. If sides a, b and angle C are given, we build the triangle as shown:
------=_NextPart_01C7E030.7DDD84F0 Content-Location: file:///C:/90F260E9/1SolvingTriangles-CosineandSinelawI_files/slide0231.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Con= struct angle C
C
Solving Triangles
       Determined Cases
SAS: Side-Angle-Sid= e
Given t= wo sides and the angles between them, it determines a triang= le. If sides a, b and angle C are given, we build the triangle as shown:
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Con= struct angle C
C
Construct
a and b
a
b
Solving Triangles
       Determined Cases
SAS: Side-Angle-Sid= e
Given t= wo sides and the angles between them, it determines a triang= le. If sides a, b and angle C are given, we build the triangle as shown:
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