Question

1. a. determine side d.

b. determine ∠e.
c. determine ∠f.

1. in a scalene triangle the measurement of the 3 angles are 34.80 degrees e and f. the sides opposite angle e and f measure 14.600 centimeters and 15.300 centimeters.

Explanation:

Part a: Determine side \( d \)

Step 1: Identify the Law of Sines

In a triangle, the Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a, b, c\) are sides opposite angles \(A, B, C\) respectively. Let's denote:

• Side opposite \(\angle E\): \(f = 14.600\) cm (wait, no, correction: Wait, the problem says "the sides opposite angle \(E\) and \(F\) measure 14.600 centimeters and 15.300 centimeters". So side opposite \(E\) is \(f = 14.600\) cm? Wait, no, let's re - label. Let's say:

Let \(\angle E\) have opposite side \(f = 14.600\) cm, \(\angle F\) have opposite side \(e = 15.300\) cm, and angle given is \(34.80^{\circ}\) (let's assume it's the third angle, but wait, the triangle is scalene, and we have two angles? Wait, no, the details say "the measurement of the 3 angles are 34.80 degrees \(E\) and \(F\)". Wait, that might be a typo. Wait, probably, one of the angles is \(34.80^{\circ}\), and we have sides opposite \(E\) and \(F\) as 14.600 and 15.300 cm. Wait, maybe the triangle has angles \(E\), \(F\), and another angle \(G = 34.80^{\circ}\), and sides opposite \(E\): \(f = 14.600\) cm, opposite \(F\): \(e = 15.300\) cm, and side \(d\) is opposite angle \(G = 34.80^{\circ}\).

Using the Law of Sines: \(\frac{d}{\sin G}=\frac{e}{\sin E}=\frac{f}{\sin F}\)

We know \(e = 15.300\) cm (opposite \(F\)), \(f = 14.600\) cm (opposite \(E\)), and \(G = 34.80^{\circ}\)

First, we can find the sum of angles in a triangle: \(E + F+G=180^{\circ}\), but we need to find \(E\) and \(F\) first? Wait, no, maybe the angle of \(34.80^{\circ}\) is one of the angles, and we have two sides. Wait, perhaps the triangle has sides: let's re - express. Let's assume that angle \(G = 34.80^{\circ}\), side opposite \(G\) is \(d\), side opposite \(E\) is \(f = 14.600\) cm, side opposite \(F\) is \(e = 15.300\) cm.

From the Law of Sines: \(\frac{d}{\sin(34.80^{\circ})}=\frac{15.300}{\sin F}=\frac{14.600}{\sin E}\)

Also, \(E + F+34.80^{\circ}=180^{\circ}\), so \(E + F = 145.2^{\circ}\)

But we can also use the Law of Sines to find the ratio of sines of \(E\) and \(F\): \(\frac{\sin E}{\sin F}=\frac{14.600}{15.300}\approx0.9542\)

Let \(E = 145.2^{\circ}-F\), then \(\sin(145.2^{\circ}-F)=\sin145.2^{\circ}\cos F-\cos145.2^{\circ}\sin F\)

\(\sin145.2^{\circ}=\sin(180 - 34.8)^{\circ}=\sin34.8^{\circ}\approx0.5709\)

\(\cos145.2^{\circ}=-\cos34.8^{\circ}\approx - 0.8214\)

So \(\sin(145.2 - F)=0.5709\cos F+0.8214\sin F\)

And \(\frac{\sin(145.2 - F)}{\sin F}=0.9542\)

\(\frac{0.5709\cos F + 0.8214\sin F}{\sin F}=0.9542\)

\(0.5709\cot F+0.8214 = 0.9542\)

\(0.5709\cot F=0.9542 - 0.8214 = 0.1328\)

\(\cot F=\frac{0.1328}{0.5709}\approx0.2326\)

\(F=\arctan(\frac{1}{0.2326})\approx77.0^{\circ}\)

Then \(E = 145.2^{\circ}-77.0^{\circ}=68.2^{\circ}\)

Now, using Law of Sines to find \(d\): \(\frac{d}{\sin(34.80^{\circ})}=\frac{15.300}{\sin(77.0^{\circ})}\)

\(\sin(77.0^{\circ})\approx0.9744\), \(\sin(34.80^{\circ})\approx0.5709\)

\(d=\frac{15.300\times\sin(34.80^{\circ})}{\sin(77.0^{\circ})}=\frac{15.300\times0.5709}{0.9744}\approx\frac{8.7348}{0.9744}\approx8.96\) cm

Step 2: Verify the calculation

We can check the Law of Sines ratios:

\(\frac{d}{\sin(34.8^{\circ})}=\frac{8.96}{0.5709}\approx15.7\)

\(\frac{15.300}{\sin(77^{\circ})}=\frac{15.300}{0.9744}\approx15.7\)

\(\frac{14.600}{\sin(68.2^{\circ})}=\frac{14.600}{0.928}\approx15.7\) (since \(\sin(68.2^{\circ})\approx0.928\))

So the calculation for \(d\) is consistent.

Part b: Determine \(\angle E\)

Step 1: Use the Law of Sines and angle sum

We know from part a that \(E + F+34.80^{\circ}=180^{\circ}\), and \(\frac{\sin E}{\sin F}=\frac{14.600}{15.300}\approx0.9542\)

Let \(E = x\), \(F = 145.2^{\circ}-x\)

\(\frac{\sin x}{\sin(145.2 - x)} = 0.9542\)

As we calculated before, \(x = E\approx68.2^{\circ}\) (we can also use the Law of Sines ratio with the found \(d\) value: \(\frac{14.600}{\sin E}=\frac{d}{\sin(34.8^{\circ})}\)

\(\sin E=\frac{14.600\times\sin(34.8^{\circ})}{d}\)

We found \(d\approx8.96\) cm, \(\sin(34.8^{\circ})\approx0.5709\)

\(\sin E=\frac{14.600\times0.5709}{8.96}=\frac{8.33514}{8.96}\approx0.930\)

\(E=\arcsin(0.930)\approx68.2^{\circ}\)

Part c: Determine \(\angle F\)

Step 1: Use angle sum property

Since the sum of angles in a triangle is \(180^{\circ}\), and we know \(E\approx68.2^{\circ}\) and the third angle \(G = 34.80^{\circ}\)

\(F=180^{\circ}-E - G\)

\(F = 180-(68.2 + 34.8)=180 - 103=77^{\circ}\) (approx, which matches our earlier calculation)

Or using Law of Sines: \(\frac{15.300}{\sin F}=\frac{d}{\sin(34.8^{\circ})}\)

\(\sin F=\frac{15.300\times\sin(34.8^{\circ})}{d}\)

\(\sin F=\frac{15.300\times0.5709}{8.96}=\frac{8.73477}{8.96}\approx0.974\)

\(F=\arcsin(0.974)\approx77^{\circ}\)

Final Answers:

a. \(d\approx\boldsymbol{8.96}\) cm (rounded to two decimal places)

b. \(\angle E\approx\boldsymbol{68.2^{\circ}}\) (rounded to one decimal place)

c. \(\angle F\approx\boldsymbol{77.0^{\circ}}\) (rounded to one decimal place)

Answer:

Step 1: Use angle sum property

Since the sum of angles in a triangle is \(180^{\circ}\), and we know \(E\approx68.2^{\circ}\) and the third angle \(G = 34.80^{\circ}\)

\(F=180^{\circ}-E - G\)

\(F = 180-(68.2 + 34.8)=180 - 103=77^{\circ}\) (approx, which matches our earlier calculation)

Or using Law of Sines: \(\frac{15.300}{\sin F}=\frac{d}{\sin(34.8^{\circ})}\)

\(\sin F=\frac{15.300\times\sin(34.8^{\circ})}{d}\)

\(\sin F=\frac{15.300\times0.5709}{8.96}=\frac{8.73477}{8.96}\approx0.974\)

\(F=\arcsin(0.974)\approx77^{\circ}\)

Final Answers:

a. \(d\approx\boldsymbol{8.96}\) cm (rounded to two decimal places)

b. \(\angle E\approx\boldsymbol{68.2^{\circ}}\) (rounded to one decimal place)

c. \(\angle F\approx\boldsymbol{77.0^{\circ}}\) (rounded to one decimal place)