Give checks to the following two questions
Use logarithms and the law of tangents to solve the triangle ABC, given that a=21.46 ft, b=46.28 ft, and C=32°28'30"
without logarithms i can do this problem easily and i have. but when i try solving it with logarithms i keep getting different answers.So i would really appreciate if you can help me show the steps with logarithms to find the solution. I know how unnecessary using logs for this problem is and if i had the choice i wouldn't because honestly its making me hate the subject in general but i have too. I would also love it if the steps were clear just so i can try doing the other problems by referring to this one.
Sure! I'll walk you through the steps to solve the triangle ABC using logarithms and the law of tangents.
Step 1: Convert the angle C from degrees, minutes, and seconds to decimal form. In this case, C = 32°28'30" becomes C = 32 + (28/60) + (30/3600) = 32.475 degrees.
Step 2: Apply the Law of Tangents to find angle A. The Law of Tangents states that (a/sin A) = (b/sin B) = (c/sin C), where a, b, and c are the sides of the triangle opposite to angles A, B, and C, respectively.
In this case, we know a = 21.46 ft, b = 46.28 ft, and C = 32.475 degrees. Let's solve for angle A using logarithms.
3. Take the logarithm (base 10) of both sides of the equation to simplify the calculations. Applying logarithms to the Law of Tangents equation, we get:
log(a/sin A) = log(b/sin B) = log(c/sin C)
4. Substitute the known values:
log(21.46/sin A) = log(46.28/sin B) = log(c/sin C)
5. Rearrange the equation to isolate sin A:
log(21.46) - log(sin A) = log(c) - log(sin C)
6. Use logarithm properties to simplify further:
log(21.46/sin A) = log(c/sin C)
log(21.46) - log(sin A) = log(c) - log(sin C)
log(21.46) - log(sin A) = log(c/sin C)
log(sin A) = log(21.46) - log(c) + log(sin C)
7. Calculate log(sin A) using a scientific calculator or logarithm table. In this case, let's assume log(sin A) = x.
log(sin A) = x
8. Solve for sin A by taking the antilogarithm of both sides of the equation:
sin A = antilog(x)
9. Finally, find angle A by taking the inverse sine (arcsine) of sin A:
A = arcsin(sin A)
Now that you have determined angle A, you can find angle B using the fact that the sum of the angles in a triangle is 180 degrees:
B = 180 - A - C
To validate your solution, you can plug the calculated values of A, B, and C back into the Law of Tangents equation and check if it holds true:
(a/sin A) = (b/sin B) = (c/sin C)
If all sides satisfy this equation, you have successfully solved the triangle ABC using logarithms and the Law of Tangents.