Question

1. determine side a .
2. in an obtuse triangle the measurement of an angle is 143 degrees 40 minutes the side opposite measures a. the measurements of the other sides are 19.000 inches and 17.500 inches.

Explanation:

Step1: Convert angle to decimal degrees

First, convert \( 143^\circ 40' \) to decimal degrees. Since \( 1^\circ = 60' \), \( 40' = \frac{40}{60} \approx 0.6667^\circ \), so \( 143^\circ 40' = 143 + 0.6667 = 143.6667^\circ \).

Step2: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \( b \), \( c \), and included angle \( A \), \( a^2 = b^2 + c^2 - 2bc \cos A \). Here, \( b = 19.000 \) in, \( c = 17.500 \) in, and \( A = 143.6667^\circ \).

First, calculate \( \cos(143.6667^\circ) \). Since \( 143.6667^\circ \) is in the second quadrant, \( \cos(143.6667^\circ) = -\cos(180^\circ - 143.6667^\circ) = -\cos(36.3333^\circ) \approx -0.8057 \).

Now, substitute into the Law of Cosines:

$$
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$$

Step3: Find the square root of \( a^2 \)

Take the square root of \( 1203.0405 \) to find \( a \):

$$ a = \sqrt{1203.0405} \approx 34.68 \text{ inches} $$

Answer:

\( \approx 34.68 \) inches