Question

law of cosines
$a^2 = b^2 + c^2 - 2bc(\cos a)$

1. determine side $a$ .

$143^\circ 40$
image of a triangle with sides 19.000 in, 17.500 in, and angle $143^\circ 40$
details

1. 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 \( 1^\circ = 60' \), \( 40'=\frac{40}{60}=\frac{2}{3}\approx0.6667^\circ \). So \( 143^\circ 40' = 143+\frac{40}{60}=143.6667^\circ \).

Step2: Apply Law of Cosines

The Law of Cosines is \( a^{2}=b^{2}+c^{2}-2bc\cos A \), where \( b = 19.000 \) in, \( c = 17.500 \) in, and \( A=143.6667^\circ \).

First, calculate \( \cos(143.6667^\circ) \). \( \cos(143.6667^\circ)=\cos(180^\circ - 36.3333^\circ)=-\cos(36.3333^\circ)\approx - 0.8057 \).

Then, calculate \( b^{2}=19.000^{2}=361 \), \( c^{2}=17.500^{2}=306.25 \), and \( 2bc = 2\times19.000\times17.500 = 665 \).

Now, substitute into the formula:

$$
LATEXBLOCK0
$$

Step3: Find \( a \)

Take the square root of \( a^{2} \): \( a=\sqrt{1203.0405}\approx34.685 \) inches.

Answer:

\( a\approx34.69 \) inches (rounded to two decimal places)