In $\Delta IJK$ , $i = 3.9$ inches, $j = 3.5$ inches and $k=2.3$ inches. Find the measure of $\angle I$ to the nearest $10$ th of a degree.

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Grade 7

Convert between customary and metric systems

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Q. In

\Delta IJK

i = 3.9

inches,

j = 3.5

inches and

k=2.3

inches. Find the measure of

\angle I

to the nearest

10

th of a degree.

Apply Law of Cosines: To find the measure of $\angle I$ , we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is $c^2 = a^2 + b^2 - 2ab\cos(C)$ , where $C$ is the angle opposite side $c$ , and $a$ and $b$ are the other two sides.
Rearrange for $\cos(I)$ : In $\Delta IJK$ , we want to find the measure of $\angle I$ , which is opposite side $i$ . Therefore, we will rearrange the Law of Cosines formula to solve for $\cos(I)$ : $\cos(I) = \frac{j^2 + k^2 - i^2}{2 \cdot j \cdot k}$ .
Plug in values: Now we plug in the values for $i$ , $j$ , and $k$ : $\cos(I) = \frac{3.5^2 + 2.3^2 - 3.9^2}{2 \times 3.5 \times 2.3}$ .
Calculate numerator: Perform the calculations inside the parentheses: $\cos(I) = \frac{12.25 + 5.29 - 15.21}{2 \times 3.5 \times 2.3}$ .
Calculate denominator: Calculate the numerator: $\cos(I) = (12.25 + 5.29 - 15.21) = 2.33$ .
Find $\cos(I)$ : Calculate the denominator: $\cos(I) = \frac{2.33}{(2 \times 3.5 \times 2.3)} = \frac{2.33}{16.1}.$
Use inverse cosine function: Divide the numerator by the denominator to find $\cos(I)$ : $\cos(I) = \frac{2.33}{16.1} \approx 0.1447$ .
Calculate inverse cosine: Now, we need to find the angle whose cosine is approximately $0.1447$ . We use the inverse cosine function to do this: $\angle I \approx \cos^{-1}(0.1447)$ .
Calculate inverse cosine: Now, we need to find the angle whose cosine is approximately $0.1447$ . We use the inverse cosine function to do this: $\angle I \approx \cos^{-1}(0.1447)$ .Use a calculator to find the inverse cosine of $0.1447$ : $\angle I \approx \cos^{-1}(0.1447) \approx 81.7$ degrees.