Expert: Sherman D. - 6/29/2009

Question
Given ΔABC with  :

1.   <A = 54° ,  b= 16 cm, and c = 18 cm.   Find a.

2.   a = 12.5 cm, b = 17.2 cm, and c = 10.6 cm. Find <C.

3.   <C = 108° ,  a = 21 cm, and  b = 24 cm.  Find c.

4.   a = 17 cm, b= 8 cm, and c = 15 cm.  Find <A.

5.   b = 28.3 cm, c = 26.5 cm and <C = 60°.  Find <A.

6.   <B = 102°,  a = 15.6 cm, and c = 15.6 cm. Find <C.

7.   <C = 56° ,  a = 34 cm, and b = 28 cm.  Find <A.


Answer
Just use the Law of Cosines or Law of Sines
a^2 = b^2 + c^2 - 2bc*cos(A)
a/sin(A) = b/sin(B) = c/sin(C)

1. <A = 54° ,  b= 16 cm, and c = 18 cm.   Find a.

a^2 = 16^2 + 18^2 - 2(16 * 18)cos(54)
a = about 15.5cm

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2. a = 12.5 cm, b = 17.2 cm, and c = 10.6 cm. Find <C.

10.6^2 = 12.5^2 + 17.2^2 - 2(12.5 * 17.2)cos(C)
C = 37°48'28.69"

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3. <C = 108° ,  a = 21 cm, and  b = 24 cm.  Find c.

c^2 = 21^2 + 24^2 - 2(21 * 24)cos(108)
c = about 36cm

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4. a = 17 cm, b= 8 cm, and c = 15 cm.  Find <A.

17^2 = 8^2 + 15^2 - 2(8 * 15)cos(A)
A = 90°

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5. b = 28.3 cm, c = 26.5 cm and <C = 60°.  Find <A.

180 - 60 = 120
A + B = 120
B = 120 - A

26.5/sin(60) = 28.3/sin(120 - A)
28.3/26.5 = sin(120 - A)/sin(60)
28.3/26.5 = (sin(120)cos(A) - sin(A)cos(120))/sin(60)

sin(120) = sin(2(60)) = 2sin(60)cos(60)
cos(120) = cos(2(60)) = 1 - 2sin(60)^2

28.3/26.5 = (2sin(60)cos(60)cos(A) - (1 - 2sin(60)^2)sin(A)))/sin(60)
28.3/26.5 = 2cos(60)cos(A) - (1/sin(60)) + 2sin(60)sin(A)
(28.3/26.5) + (1/sin(60)) = 2cos(60)cos(A) + 2sin(60)sin(A)
(28.3sin(60) + 26.5)/(53sin(60)) = cos(60 - A)
A = Invalid

I've done this the long way and i've done this the short way, but my answer is always the same, that its invalid as in it isn't possible.

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6. <B = 102°,  a = 15.6 cm, and c = 15.6 cm. Find <C.

seeing as how "a" and "c" are equal, this would be an isosceles triangle and therefore the angles opposite of "a" and "b" are equal

(180 - 102)/2 = 39

Angle C is 39°

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7. <C = 56° ,  a = 34 cm, and b = 28 cm.  Find <A.

180 - 56 = 124
A + B = 124
A = 124 - B

34/sin(A) = 28/sin(B)

34/sin(124 - B) = 28/sin(B)
34/28 = sin(124 - B)/sin(B)
34/28 = (sin(124)cos(B) - sin(B)cos(124))/sin(B)
17/14 = ((sin(124)cos(B))/sin(B)) - cos(124)
(17/14) + cos(124) = sin(124)cos(B)/sin(B)
1/((17/14) + cos(124)) = sin(B)/(sin(124)cos(B))
sin(B)/cos(B) = sin(124)/((17/14) + cos(124))
tan(B) = sin(124)/((17/14) + cos(124))
B = about 51°41'5.0959 or about 52°

A = about 72°18'54.9

info found at http://www.clarku.edu/~djoyce/trig/laws.html