In ΔIJK, i = 3.3 inches, k = 7 inches and ∠K=117°. Find all possible values of ∠I, to the nearest 10th of a degree.

To solve this problem, we will use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle γ opposite side c, the following relationship holds:

c² = a² + b² - 2ab cos(γ)

In this case, we know the lengths of sides i and k, and the measure of ∠K. We can rearrange the Law of Cosines to solve for cos(γ):

cos(γ) = (a² + b² - c²) / (2ab)

Substituting the given values:

cos(∠I) = (i² + k² - j²) / (2ik) cos(∠I) = (3.3² + 7² - j²) / (2 * 3.3 * 7)

We don't know the length of side j, but we do know that ∠K = 117°. We can use the Law of Sines to find j:

sin(∠K) = j / k j = k * sin(∠K) j = 7 * sin(117°)

Substitute j back into the equation for cos(∠I):

cos(∠I) = (3.3² + 7² - (7 * sin(117°))²) / (2 * 3.3 * 7)

Solve for ∠I:

∠I = arccos[((3.3² + 7² - (7 * sin(117°))²) / (2 * 3.3 * 7))]

This will give you the measure of ∠I in degrees. Note that because the arccos function returns values between 0° and 180°, there may be two possible values for ∠I. To find the second possible value, subtract the first value from 180°.