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Math Help - The Ambiguous Case...Two Solutions

  1. #1
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    Smile The Ambiguous Case...Two Solutions

    Solve the triangle (Two Solutions)

    a = 6, b = 8, alpha = 35 degrees
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  2. #2
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    Quote Originally Posted by magentarita View Post
    Solve the triangle (Two Solutions)

    a = 6, b = 8, alpha = 35 degrees
    Is alpha the angle opposite side c or opposite one of the other sides (side a or side b)? And what are you trying to find ..... side c, or the other two angles, or side c and the other two angles?

    Your post is well named
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  3. #3
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    Quote Originally Posted by magentarita View Post
    Solve the triangle (Two Solutions)

    a = 6, b = 8, alpha = 35 degrees
    here you see, a<b, so calculate, b \sin \alpha

    \alpha= A,

    b \sin \alpha= 8\times \sin 35

    =4.5<6

    a > b \sin \alpha

    so, two solutions, two triangles. Draw the diagram. CB = CB' = 6 cm = a
    we will use two triangles ABC and AB'C

    use sine law in triangle AB'Cto find angle B'

    \frac{8}{\sin B'}= \frac{6}{\sin \alpha}

    \frac{8}{\sin B'}= \frac{6}{\sin 35}

    6 \sin B' = 8 \sin 35

    sin B' = 0.764768581

    B'= 49.88 = 50 degrees.

    In triangle CBB', angle B = angle B' = 50 degrees.

    So, supplementary angle B = 180 - 50 = 130 degrees.

    In triangle ACB, angle C = 180 - (35 + 130) = 15 degrees.

    In triangle ACB', angle C = 180 - (35 + 50) = 95 degrees.

    Use sin law again in triangle ACB,

    \frac {AB}{\sin 15}=\frac{6}{\sin 35}

    AB = 2.7 cm

    Use sin law again in triangle ACB',

    \frac {AB'}{\sin 95}=\frac{6}{\sin 35}

    AB = 10.4 cm

    Solution is :

    For triangle ABC,
    angle A= \alpha = 35 degrees, B = 130 degrees, C = 15 degrees
    a= 6 cm, b = 8 cm, c = AB = 2.7 cm

    For triangle AB'C,
    angle A= \alpha = 35 degrees, B' = 50 degrees, C = 95 degrees
    a= 6 cm, b = 8 cm, c = AB' = 10.4 cm
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  4. #4
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    look...

    Quote Originally Posted by mr fantastic View Post
    Is alpha the angle opposite side c or opposite one of the other sides (side a or side b)? And what are you trying to find ..... side c, or the other two angles, or side c and the other two angles?

    Your post is well named
    Alpha = side a

    A for a, get it?
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  5. #5
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    Perfect

    Quote Originally Posted by Shyam View Post
    here you see, a<b, so calculate, b \sin \alpha

    \alpha= A,

    b \sin \alpha= 8\times \sin 35

    =4.5<6

    a > b \sin \alpha

    so, two solutions, two triangles. Draw the diagram. CB = CB' = 6 cm = a
    we will use two triangles ABC and AB'C

    use sine law in triangle AB'Cto find angle B'

    \frac{8}{\sin B'}= \frac{6}{\sin \alpha}

    \frac{8}{\sin B'}= \frac{6}{\sin 35}

    6 \sin B' = 8 \sin 35

    sin B' = 0.764768581

    B'= 49.88 = 50 degrees.

    In triangle CBB', angle B = angle B' = 50 degrees.

    So, supplementary angle B = 180 - 50 = 130 degrees.

    In triangle ACB, angle C = 180 - (35 + 130) = 15 degrees.

    In triangle ACB', angle C = 180 - (35 + 50) = 95 degrees.

    Use sin law again in triangle ACB,

    \frac {AB}{\sin 15}=\frac{6}{\sin 35}

    AB = 2.7 cm

    Use sin law again in triangle ACB',

    \frac {AB'}{\sin 95}=\frac{6}{\sin 35}

    AB = 10.4 cm

    Solution is :

    For triangle ABC,
    angle A= \alpha = 35 degrees, B = 130 degrees, C = 15 degrees
    a= 6 cm, b = 8 cm, c = AB = 2.7 cm

    For triangle AB'C,
    angle A= \alpha = 35 degrees, B' = 50 degrees, C = 95 degrees
    a= 6 cm, b = 8 cm, c = AB' = 10.4 cm
    Perfectly done!
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  6. #6
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    Quote Originally Posted by magentarita View Post
    Alpha = side a

    A for a, get it?
    1. Clarity when asking your questions is not too much to ask for.

    2. When you're getting free professional help from someone it's wise not to get too cute with them. You cut them a bit of slack when they ask you a question.

    Get it?
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  7. #7
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    Quote Originally Posted by magentarita View Post
    Alpha = side a

    A for a, get it?
    Alpha is the angle opposite side a.

    Show some respect for others trying to help by making your statements clear, and avoid potentialy offensive comments like: "A for a, get it?". If someone more experienced with this stuff has trouble understanding what you write perhaps you should consider that the fault may lie with you rather than with them.

    RonL
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