Question

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in $\triangle hij$, $i = 7.3$ inches, $h = 6.4$ inches and $\angle h = 57^\circ$. find all possible values of $\angle i$, to the nearest 10th of a degree.
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Explanation:

Step1: Apply the Law of Sines

We use the Law of Sines, which states that $\frac{\sin H}{h}=\frac{\sin I}{i}$. Given $H = 57^\circ$, $h = 6.4$ inches, and $i = 7.3$ inches. Plugging in the values, we get $\frac{\sin 57^\circ}{6.4}=\frac{\sin I}{7.3}$.

Step2: Solve for $\sin I$

First, calculate $\sin 57^\circ\approx0.8387$. Then, cross - multiply: $\sin I=\frac{7.3\times\sin 57^\circ}{6.4}$. Substitute the value of $\sin 57^\circ$: $\sin I=\frac{7.3\times0.8387}{6.4}$. Calculate the numerator: $7.3\times0.8387 = 6.12251$. Then, $\sin I=\frac{6.12251}{6.4}\approx0.9566$.

Step3: Find the possible angles for $I$

Since $\sin I\approx0.9566$, we know that $I=\sin^{- 1}(0.9566)\approx73.1^\circ$ or $I = 180^\circ - 73.1^\circ=106.9^\circ$. We need to check if these angles are valid in the triangle. The sum of angles in a triangle is $180^\circ$. For $I = 73.1^\circ$, the third angle $J=180^\circ-(57^\circ + 73.1^\circ)=49.9^\circ$, which is valid. For $I = 106.9^\circ$, the third angle $J=180^\circ-(57^\circ+106.9^\circ)=16.1^\circ$, which is also valid. But we need to check the side lengths. Since $i>h$, angle $I$ can be acute or obtuse? Wait, actually, when we use the Law of Sines, if $i>h$, then angle $I$ can be either acute or obtuse? Wait, no, when $i > h$, then $I$ must be greater than $H$. But let's check the sine value. Since $\sin I\approx0.9566$, the reference angle is $\approx73.1^\circ$, and the other angle is $180 - 73.1=106.9^\circ$. Now, we check if $I = 106.9^\circ$ is possible. The sum of $H$ and $I$ should be less than $180^\circ$. $57^\circ+106.9^\circ = 163.9^\circ<180^\circ$, so it is possible. But wait, let's re - examine the Law of Sines. The formula is $\frac{\sin H}{h}=\frac{\sin I}{i}$. Since $i>h$, then $\sin I>\sin H$. Since $H = 57^\circ$, $\sin H\approx0.8387$, and $\sin I\approx0.9566>0.8387$, so there are two possible triangles? Wait, no, the ambiguous case (SSA) occurs when $h < i$ and $h>i\sin H$. Let's calculate $i\sin H=7.3\times\sin 57^\circ\approx7.3\times0.8387\approx6.122$. Since $h = 6.4>6.122$, so there are two possible triangles. But the question says "Find all possible values of $\angle I$". So the possible values are approximately $73.1^\circ$ and $106.9^\circ$. But let's check the calculation again. Wait, when we calculated $\sin I=\frac{7.3\times\sin 57^\circ}{6.4}$, let's recalculate:

$\sin 57^\circ\approx0.8386705679454239$

$7.3\times0.8386705679454239 = 7.3\times0.8386705679454239\approx6.1223$

$\frac{6.1223}{6.4}\approx0.9566$

$\sin^{-1}(0.9566)\approx73.1^\circ$ (using a calculator). The other angle is $180 - 73.1 = 106.9^\circ$. Now, we need to check if $I = 106.9^\circ$ is valid. The sum of angles: $57+106.9 = 163.9<180$, so it is valid. But wait, the problem says "to the nearest tenth of a degree". Let's check the initial side lengths. Since $i = 7.3$ and $h = 6.4$, and $H = 57^\circ$. So the two possible angles for $I$ are approximately $73.1^\circ$ and $106.9^\circ$. But let's see, maybe I made a mistake in the ambiguous case. Wait, the formula for the ambiguous case: if $h < i$ and $h>i\sin H$, then two triangles exist. Here, $i\sin H=7.3\times\sin57^\circ\approx6.12$, and $h = 6.4>6.12$, so two triangles exist, so two possible angles for $I$. But let's check with the calculator for $\sin^{-1}(0.9566)$. Using a calculator, $\sin^{-1}(0.9566)\approx73.1^\circ$, and the supplementary angle is $180 - 73.1 = 106.9^\circ$.

Answer:

The possible values of $\angle I$ are $73.1^\circ$ and $106.9^\circ$. If we consider the options (Additional Angle or No Possible Angles), since there are two possible angles, the answer is $73.1$ (and also $106.9$, but if we take the acute one first) or considering the problem's "Additional Angle" option, the additional angle is $106.9^\circ$ and the first angle is $73.1^\circ$. But if we follow the calculation, the two possible angles are approximately $73.1^\circ$ and $106.9^\circ$.