Solve the triangle. Round to the nearest tenth.
B=54deg a=42 c=6.5
1. I found length of b using law of cosine and got 36.2
2. Using side of b, I used the law of sine to get angle A = 28.2
3. I added B+A and subtracted from 180 to get angle C = 127.8 which DOES NOT make sense since side c is the smallest side. I can't seem to find my error. Help please?
Thanks
b^2 = a^2 + c^2 - 2ac cos 54 = 1485
b = 38.5
Check your numbers again.
I get sinA = a*sinB/b = .1366
A = 7.85 degrees
I didn't get A = 7.85 degs...
I got 61.8deg for A this time... which would make C have 64.2deg which still doesn't make sense...
You are correct; I inserted the wrong value of a in the law of sines. Sin A = 0.88256 and A = 62 degrees or 118 degrees. You have to choose the larger value. That means C = 180 - 118 - 54 = 8 degrees
It seems like you made a mistake in step 3 of your calculations. Let's go through the steps again to find the correct value for angle C.
1. You correctly used the Law of Cosines to find the length of side b. Given that a = 42, c = 6.5, and B = 54 degrees, you can use the formula:
b² = a² + c² - 2ac * cos(B)
Plugging in the values,
b² = 42² + 6.5² - 2 * 42 * 6.5 * cos(54)
b² = 1764 + 42.25 - 546 * 0.5878
b² ≈ 1845.57
Taking the square root, b ≈ 42.9 (rounded to the nearest tenth).
2. Now, let's find angle A. We can use the Law of Sines to relate the sides and angles:
sin(A) / a = sin(B) / b
Plugging in the values,
sin(A) / 42 ≈ sin(54) / 42.9
Rearranging the equation and solving for sin(A),
sin(A) ≈ (sin(54) / 42.9) * 42
sin(A) ≈ 0.80902
Now, we can take the inverse sine to find A,
A ≈ arcsin(0.80902)
A ≈ 53.2 (rounded to the nearest tenth).
3. Finally, to find angle C, we can use the fact that the sum of angles in a triangle is 180 degrees:
C = 180 - A - B
C = 180 - 53.2 - 54
C ≈ 72.8 (rounded to the nearest tenth).
Therefore, the correct answer is:
A ≈ 53.2 degrees
B ≈ 54 degrees
C ≈ 72.8 degrees
If you double-check your calculations using these steps, it should help you identify the error you made.