Question

1. a. determine side ( d ).

b. determine ( angle e ).
c. determine ( angle f ).

1. in an obtuse triangle the measurement of 3 angles are 107 degrees 0 minutes, ( e ) and ( f ). the measurements of the side opposite angle ( f ) is 3.700 inches and opposite angle ( e ) is 4.300 inches.

Explanation:

Part a: Determine side \( d \)

Step 1: Identify the triangle type and given values

We have an obtuse triangle with one angle \( 107^\circ 0' \) (let's call this angle \( G \) for clarity), side opposite \( \angle F \) is \( f = 3.700 \) inches, side opposite \( \angle E \) is \( e = 4.300 \) inches, and we need to find side \( d \) (opposite \( \angle G = 107^\circ \)). We can use the Law of Cosines: \( d^2 = e^2 + f^2 - 2ef \cos G \).

Step 2: Convert angle to decimal degrees

\( 107^\circ 0' = 107^\circ \). \( \cos(107^\circ) \approx \cos(180^\circ - 73^\circ)= -\cos(73^\circ) \approx -0.2924 \).

Step 3: Substitute values into Law of Cosines

\( e = 4.300 \), \( f = 3.700 \), \( \cos G \approx -0.2924 \)

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Part b: Determine \( \angle E \)

Step 1: Use Law of Sines

Law of Sines: \( \frac{\sin E}{f}=\frac{\sin G}{d} \)
We know \( f = 3.700 \), \( G = 107^\circ \), \( d\approx6.44 \)

$$ \sin E=\frac{f\sin G}{d}=\frac{3.700\times\sin(107^\circ)}{6.44} $$

\( \sin(107^\circ)=\sin(73^\circ)\approx0.9563 \)

$$ \sin E=\frac{3.700\times0.9563}{6.44}\approx\frac{3.5383}{6.44}\approx0.5494 $$

Step 2: Find \( \angle E \)

\( \angle E=\arcsin(0.5494)\approx33.3^\circ \) (since triangle is obtuse at \( G \), \( E \) must be acute)

Part c: Determine \( \angle F \)

Step 1: Use angle sum property

Sum of angles in triangle: \( E + F + G = 180^\circ \)
\( F = 180^\circ - E - G \)
\( E\approx33.3^\circ \), \( G = 107^\circ \)
\( F = 180 - 33.3 - 107 = 39.7^\circ \) (we can also verify with Law of Sines: \( \frac{\sin F}{e}=\frac{\sin G}{d} \), \( \sin F=\frac{4.300\times\sin(107^\circ)}{6.44}\approx\frac{4.3\times0.9563}{6.44}\approx\frac{4.112}{6.44}\approx0.638 \), \( F=\arcsin(0.638)\approx39.6^\circ \), consistent with angle sum)

Answer:

s:
a. \( d\approx\boldsymbol{6.44} \) inches (approximate)
b. \( \angle E\approx\boldsymbol{33.3^\circ} \) (approximate)
c. \( \angle F\approx\boldsymbol{39.7^\circ} \) (approximate)