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: Recall the Law of Sines

The Law of Sines states that for a triangle with sides \( a, b, c \) opposite angles \( A, B, C \) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In our triangle, we know one angle (let's assume the given \( 34.80^\circ \) is an angle, say angle opposite to side \( d \)? Wait, no, looking at the diagram and the details: the side opposite angle \( E \) is \( 14.600 \) cm, opposite angle \( F \) is \( 15.300 \) cm, and there is an angle of \( 34.80^\circ \). Wait, maybe the angle given is, let's re - check. The Law of Sines formula given is \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}=\frac{14.600}{\sin E}\). So we can use the Law of Sines to find \( d \). Let's assume that the angle \( 34.80^\circ \) is an angle (let's say angle \( D \), and side \( d \) is opposite angle \( D = 34.80^\circ \), side opposite \( E \) is \( 14.600 \) cm, side opposite \( F \) is \( 15.300 \) cm.

From the Law of Sines: \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\) or \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}\). But we can also use the two - known sides and the Law of Sines relation. Wait, first, let's find angle \( E \) or \( F \) first? Wait, no, for part a, to find \( d \), we can use the Law of Sines. Let's assume that we can use the ratio \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we need to find \(\sin E\) first? Wait, no, maybe the angle \( 34.80^\circ \) is related to the triangle. Wait, the sum of angles in a triangle is \( 180^\circ \). Wait, maybe there is a typo in the initial writing, and the angle is \( 34.80^\circ \), and we can use the Law of Sines as \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we can also use the formula \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}\). Wait, let's calculate \(\sin(34.80^\circ)\approx\sin(34.8^{\circ})\approx0.5707\) (using calculator: \( \sin(34.8)=\sin(34 + 0.8)=\sin34\cos0.8+\cos34\sin0.8\approx0.5592\times0.9999 + 0.8290\times0.01396\approx0.5592+0.0115\approx0.5707\)).

From the Law of Sines, \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we can also use \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}\). Wait, maybe the angle \( 34.80^\circ \) is angle \( D \), and side \( d \) is opposite angle \( D \), side opposite \( E \) is \( 14.600 \), side opposite \( F \) is \( 15.300 \). Let's use the Law of Sines: \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we can first find angle \( E \) from part b? Wait, no, let's do part a first. Wait, maybe the given formula is \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}=\frac{14.600}{\sin E}\). Let's assume that we can write \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we can also use the fact that \(\frac{14.600}{\sin E}=\frac{15.300}{\sin F}\), and \( E + F+34.80^\circ = 180^\circ\), so \( E + F=145.2^\circ\).

But for part a, let's use the Law of Sines. Let's take \(\frac{d}{\sin(34.80^\circ)}=\frac{14.600}{\sin E}\), but we can also calculate \( d \) as follows: from \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}\), but we need another relation. Wait, maybe the angle \( 34.80^\circ \) is the angle opposite to side \( d \), and we have two other sides: \( 14.600 \) (opposite \( E \)) and \( 15.300 \) (opposite \( F \)). So using the Law of Sines:

\(d=\frac{14.600\times\sin(34.80^\circ)}{\sin E}\), but we can find \(\sin E\) from \(\frac{\sin E}{14.600}=\frac{\sin F}{15.300}\), and \( E + F = 180 - 34.8=145.2^\circ\). Let \( E=x\), then \( F = 145.2 - x\). So \(\frac{\sin x}{14.6}=\frac{\sin(145.2 - x)}{15.3}\). Using the sine subtraction formula: \(\sin(145.2 - x)=\sin145.2\cos x-\cos145.2\sin x\). \(\sin145.2=\sin(180 - 34.8)=\sin34.8\approx0.5707\), \(\cos145.2=\cos(180 - 34.8)=-\cos34.8\approx - 0.8214\).

So \(\frac{\sin x}{14.6}=\frac{0.5707\cos x+0.8214\sin x}{15.3}\)

Multiply both sides by \(14.6\times15.3\):

\(15.3\sin x=14.6(0.5707\cos x + 0.8214\sin x)\)

\(15.3\sin x=8.3322\cos x+11.9924\sin x\)

\(15.3\sin x - 11.9924\sin x=8.3322\cos x\)

\(3.3076\sin x=8.3322\cos x\)

\(\tan x=\frac{8.3322}{3.3076}\approx2.519\)

\(x = \arctan(2.519)\approx68.3^\circ\) (this is angle \( E \), which we can check in part b)

Now, for part a, \(d=\frac{14.600\times\sin(34.80^\circ)}{\sin E}\). \(\sin(34.80^\circ)\approx0.5707\), \(\sin E=\sin(68.3^\circ)\approx0.929\)

\(d=\frac{14.600\times0.5707}{0.929}\approx\frac{8.3322}{0.929}\approx8.97\) cm (approximate value, let's check with more precise calculation)

Wait, maybe a better way: from the Law of Sines, \(\frac{d}{\sin(34.80^\circ)}=\frac{15.300}{\sin F}\), and \( F = 180 - 34.8 - 68.3=76.9^\circ\), \(\sin F=\sin(76.9^\circ)\approx0.972\)

\(d=\frac{15.300\times\sin(34.80^\circ)}{\sin F}=\frac{15.300\times0.5707}{0.972}\approx\frac{8.7317}{0.972}\approx8.98\) cm. Let's use more precise values.

\(\sin(34.8^\circ)\): using calculator, \(34.8^\circ\) in radians is \(34.8\times\frac{\pi}{180}\approx0.607\) radians. \(\sin(0.607)\approx0.5706\)

\(\sin(68.3^\circ)\): \(68.3\times\frac{\pi}{180}\approx1.192\) radians, \(\sin(1.192)\approx0.929\)

\(d=\frac{14.6\times0.5706}{0.929}=\frac{8.33076}{0.929}\approx8.97\) cm.

Part b: Determine \( \angle E \)

Step 1: Use the Law of Sines

We know that \(\frac{\sin E}{14.600}=\frac{\sin F}{15.300}\), and \( E + F=180 - 34.8 = 145.2^\circ\), so \( F = 145.2 - E\)

Substitute into the Law of Sines: \(\frac{\sin E}{14.6}=\frac{\sin(145.2 - E)}{15.3}\)

As we did before, \(\sin(145.2 - E)=\sin145.2\cos E-\cos145.2\sin E\)

\(\sin145.2=\sin(180 - 34.8)=\sin34.8\approx0.5706\)

\(\cos145.2=\cos(180 - 34.8)=-\cos34.8\approx - 0.8214\)

So \(\frac{\sin E}{14.6}=\frac{0.5706\cos E + 0.8214\sin E}{15.3}\)

Cross - multiply: \(15.3\sin E=14.6(0.5706\cos E + 0.8214\sin E)\)

\(15.3\sin E=8.33076\cos E+11.99244\sin E\)

\(15.3\sin E-11.99244\sin E = 8.33076\cos E\)

\(3.30756\sin E=8.33076\cos E\)

\(\tan E=\frac{8.33076}{3.30756}\approx2.519\)

Step 2: Find \( E \)

\(E=\arctan(2.519)\approx68.3^\circ\) (we can check with calculator: \(\arctan(2.519)\approx68.3^\circ\))

Part c: Determine \( \angle F \)

Step 1: Use the angle sum property

The sum of angles in a triangle is \(180^\circ\). We know one angle is \(34.80^\circ\) and we found \( \angle E\approx68.3^\circ\)

Step 2: Calculate \( \angle F \)

\(\angle F=180-(34.80 + 68.3)=180 - 103.1 = 76.9^\circ\) (or using the Law of Sines: \(\sin F=\frac{15.300\times\sin(34.80^\circ)}{d}\), and since \(d\approx8.97\), \(\sin F=\frac{15.3\times0.5706}{8.97}=\frac{8.73018}{8.97}\approx0.973\), so \(F=\arcsin(0.973)\approx76.9^\circ\))

Answers:

a. \(d\approx\boldsymbol{8.97}\) cm (or more precisely, depending on calculation, around 8.97 - 8.98 cm)

b. \(\angle E\approx\boldsymbol{68.3^\circ}\) (or using more precise calculation, around \(68.3^\circ\))

c. \(\angle F\approx\boldsymbol{76.9^\circ}\) (or around \(76.9^\circ\))

Answer:

s:
a. \(d\approx\boldsymbol{8.97}\) cm (or more precisely, depending on calculation, around 8.97 - 8.98 cm)

b. \(\angle E\approx\boldsymbol{68.3^\circ}\) (or using more precise calculation, around \(68.3^\circ\))

c. \(\angle F\approx\boldsymbol{76.9^\circ}\) (or around \(76.9^\circ\))