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Find the lengths of the missing sides if side [tex]a[/tex] is opposite angle [tex]A[/tex], side [tex]b[/tex] is opposite angle [tex]B[/tex], and side [tex]C[/tex] is the hypotenuse.

Given:
[tex]A = \frac{.5}{12}[/tex], [tex]b = 6[/tex]

Possible solutions:
- [tex]a = 2, \, c = 6[/tex]
- [tex]a = 2.5, \, c = 6.5[/tex]
- [tex]a = 3, \, c = 7[/tex]
- [tex]a = 6.5, \, c = 2.5[/tex]



Answer :

GinnyAnswer
To determine if the given sets of side lengths form a right triangle with the angle [tex]\( A \)[/tex] opposite side [tex]\( a \)[/tex] and hypotenuse [tex]\( c \)[/tex], we use the trigonometric relationship involving the cosine of the angle. Specifically, the cosine of an angle [tex]\( A \)[/tex] in a right triangle is given by:

[tex]\[ \cos(A) = \frac{a}{c} \][/tex]

We know the angle [tex]\( A \)[/tex] is given as [tex]\( \frac{0.5}{12} \)[/tex].

1. Calculate the cosine of angle [tex]\( A \)[/tex]:
[tex]\[ \cos(A) = \frac{0.5}{12} \approx 0.04166667 \][/tex]

Now, we will verify each pair of provided values ([tex]\( a \)[/tex] and [tex]\( c \)[/tex]) to see if they yield the same cosine value.

### Case 1: [tex]\( a = 2 \)[/tex], [tex]\( c = 6 \)[/tex]

[tex]\[ \cos(A) = \frac{2}{6} = \frac{1}{3} \approx 0.333333 \][/tex]

This does not equal [tex]\( 0.04166667 \)[/tex].

### Case 2: [tex]\( a = 2.5 \)[/tex], [tex]\( c = 6.5 \)[/tex]

[tex]\[ \cos(A) = \frac{2.5}{6.5} \approx 0.384615 \][/tex]

This does not equal [tex]\( 0.04166667 \)[/tex].

### Case 3: [tex]\( a = 3 \)[/tex], [tex]\( c = 7 \)[/tex]

[tex]\[ \cos(A) = \frac{3}{7} \approx 0.428571 \][/tex]

This does not equal [tex]\( 0.04166667 \)[/tex].

### Case 4: [tex]\( a = 6.5 \)[/tex], [tex]\( c = 2.5 \)[/tex]

[tex]\[ \cos(A) = \frac{6.5}{2.5} = 2.6 \][/tex]

This does not equal [tex]\( 0.04166667 \)[/tex].

None of these sets of values meet the given condition. Therefore, there are no pairs of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] among the provided sets that form the desired angle [tex]\( A \)[/tex]. The resulting list of possible values that satisfy the given condition is:

[tex]\[ [] \][/tex]

This indicates that none of the given side length pairs make the angle [tex]\( A \)[/tex] equal to [tex]\( \frac{0.5}{12} \)[/tex].

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