In /_\HIJ,j=8.6cm,i=4.6cm and /_I=158^(@). Find all possible values of /_J, to the nearest 10th of a degree. Answer:

Use Law of Sines: Use the Law of Sines to find the possible values of angle J. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. That is, for triangle HIJ: sin(I)/i = sin(J)/j We can rearrange this to solve for sin(J): sin(J) = (sin(I) * j) / i First, we need to calculate sin(I). Since I = 158 degrees: sin(I) = sin(158°)Calculate sin(I): Calculate sin(I) using a calculator. sin(158°) ≈ 0.309Substitute values for sin(J): Substitute the values of sin(I), j, and i into the equation to find sin(J). sin(J) = (0.309 * 8.6 cm) / 4.6 cmPerform sin(J) calculation: Perform the calculation to find sin(J). sin(J) ≈ (0.309 * 8.6) / 4.6 sin(J) ≈ 2.6574 / 4.6 sin(J) ≈ 0.5777Find possible values of J: Find the possible values of angle J by taking the inverse sine (arcsin) of sin(J). Since the sine function is positive in the first and second quadrants, there could be two possible angles for J, one acute and one obtuse. However, since we are dealing with a triangle, the sum of angles I and J must be less than 180 degrees. Given that angle I is already 158 degrees, angle J must be acute. J ≈ arcsin(0.5777)Calculate angle J: Calculate the value of angle J using a calculator. J ≈ arcsin(0.5777) J ≈ 35.3 degrees (to the nearest tenth)

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Q. In

\triangle \mathrm{HIJ}, j=8.6 \mathrm{~cm}, i=4.6 \mathrm{~cm}

and

\angle \mathrm{I}=158^{\circ}

. Find all possible values of

\angle \mathrm{J}

, to the nearest

10

th of a degree.

\newline

Answer:

Use Law of Sines: Use the Law of Sines to find the possible values of angle J. $\newline$ The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. That is, for triangle HIJ: $\newline$ $\frac{\sin(I)}{i} = \frac{\sin(J)}{j}$ $\newline$ We can rearrange this to solve for $\sin(J)$ : $\newline$ $\sin(J) = \frac{\sin(I) \cdot j}{i}$ $\newline$ First, we need to calculate $\sin(I)$ . Since $I = 158$ degrees: $\newline$ $\sin(I) = \sin(158^\circ)$
Calculate $\sin(I)$ : Calculate $\sin(I)$ using a calculator. $\sin(158°) \approx 0.309$
Substitute values for sin(J): Substitute the values of $sin(I)$ , $j$ , and $i$ into the equation to find $sin(J)$ . $\sin(J) = \frac{0.309 \times 8.6\,\text{cm}}{4.6\,\text{cm}}$
Perform $\sin(J)$ calculation: Perform the calculation to find $\sin(J)$ . $\newline$ $\sin(J) \approx (0.309 \times 8.6) / 4.6$ $\newline$ $\sin(J) \approx 2.6574 / 4.6$ $\newline$ $\sin(J) \approx 0.5777$
Find possible values of $J$ : Find the possible values of angle $J$ by taking the inverse sine (arcsin) of $ext{sin}(J)$ . $\newline$ Since the sine function is positive in the first and second quadrants, there could be two possible angles for $J$ , one acute and one obtuse. However, since we are dealing with a triangle, the sum of angles $I$ and $J$ must be less than $180$ degrees. Given that angle $I$ is already $158$ degrees, angle $J$ must be acute. $\newline$ $J$ $0$
Calculate angle J: Calculate the value of angle J using a calculator. $\newline$ $J \approx \arcsin(0.5777)$ $\newline$ $J \approx 35.3$ degrees (to the nearest tenth)