A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $1$ $1$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what is the length of side A?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $1$ $1$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what is the length of side A?

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Answer 1 · 2016-05-02T14:08:14

$A \approx 0.3660 to 4 decimal places$ $color(blue)(A~~0.3660" to 4 decimal places")$

Explanation:

Good practice to draw a diagram so that you can see what is going on.
Tony B

It looks as though things are changing! I always understood that capital letters stood for the vertices (angles) and lower case was for the sides.

Momentarily using the notation I am used to in that Capital letters represent vertices:

Using the sine rule $\frac{a}{sin (A)} = \frac{b}{sin (B)} = \frac{c}{sin (C)}$ $" " a/(sin(A))=b/(sin(B))=c/(sin(C))$
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Using your notation:

$\frac{C}{sin (\frac{3}{4} π)} = \frac{A}{sin (\frac{π}{12})}$ $" "C/(sin(3/4 pi))=A/(sin(pi/12))$

But $C = 1$ $C=1$ giving:

$\frac{1}{sin (\frac{3}{4} π)} = \frac{A}{sin (\frac{π}{12})}$ $" "1/(sin(3/4 pi))=A/(sin(pi/12))$

Multiply both sides by $sin (\frac{π}{12})$ $sin(pi/12)$

$\frac{sin (\frac{π}{12})}{sin (\frac{3}{4} π)} = A$ $" "(sin(pi/12))/(sin(3/4 pi))=A$

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Note that $\frac{1}{12} π = \frac{1}{12} \times 180 = 15^{o}$ $1/12 pi = 1/12xx180 = 15^o$

Note that $\frac{3}{4} π = \frac{3}{4} \times 180 = 135^{o}$ $3/4pi=3/4xx180=135^o$
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$A \approx 0.3660 to 4 decimal places$ $color(blue)(A~~0.3660" to 4 decimal places")$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $1$ $1$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what is the length of side A?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $1$ $1$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what is the length of side A?