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$In /_\KLM,k=550cm,m=540cm and /_M=74^(@). Find all possible values of /_K, to the nearest 1oth of a degree. Answer:$

In $\triangle \mathrm{KLM}, k=550 \mathrm{~cm}, m=540 \mathrm{~cm}$ and $\angle \mathrm{M}=74^{\circ}$ . Find all possible values of $\angle \mathrm{K}$ , to the nearest $10$ th of a degree. $\newline$ Answer:

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Q. In

\triangle \mathrm{KLM}, k=550 \mathrm{~cm}, m=540 \mathrm{~cm}

and

\angle \mathrm{M}=74^{\circ}

. Find all possible values of

\angle \mathrm{K}

, to the nearest

10

th of a degree.

\newline

Answer:

Given triangle KLM: We are given a triangle KLM with sides $k=550\,\text{cm}$ , $m=540\,\text{cm}$ , and angle $M=74^\circ$ . We want to find the possible values of angle $K$ . We can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Law of Sines: First, we write the Law of Sines for our triangle: $(\sin(K)/m) = (\sin(M)/k)$
Write Law of Sines: We plug in the known values: $(\sin(K)/540) = (\sin(74^\circ)/550)$
Plug in values: Now we solve for $\sin(K)$ : $\sin(K) = \left(\frac{\sin(74^\circ)}{550}\right) \cdot 540$
Solve for $\sin(K)$ : We calculate the value of $\sin(74^\circ)$ using a calculator: $\newline$ $\sin(74^\circ) \approx 0.9613$
Calculate $\sin(74^\circ)$ : We substitute this value into our equation: $\newline$ $\sin(K) = \frac{0.9613}{550} \times 540$
Substitute value: We perform the calculation: $\sin(K) \approx (\frac{0.9613}{550}) \times 540 \approx 0.9419$
Perform calculation: Now we find the angle $K$ whose sine is approximately $0.9419$ . We use the inverse sine function (arcsin) on a calculator: $\newline$ $K \approx \arcsin(0.9419)$
Find angle $K$ : We calculate the value of $K$ : $K \approx 70.5$ degrees
Calculate value of extit{K}: However, since the sine function is positive in both the first and second quadrants, there is another possible value for angle extit{K}. It is the supplement of extit{ $70$ . $5$ } degrees, which is extit{ $180$ } degrees - extit{ $70$ . $5$ } degrees.
Find supplement of K: We calculate the supplement of K: $\newline$ $K \approx 180$ degrees $- 70.5$ degrees $\approx 109.5$ degrees
Calculate supplement of K: We now have two possible values for angle $K$ : $70.5$ degrees and $109.5$ degrees. However, we must check if these values are valid by ensuring that the sum of angles in a triangle is $180$ degrees.
Check validity of values: We add the known angle $M$ and both possible values of angle $K$ to see if they sum to $180$ degrees: $74$ degrees + $70.5$ degrees + angle $L$ = $180$ degrees $74$ degrees + $109.5$ degrees + angle $L$ = $180$ degrees
Add angles: We solve for angle $L$ in both cases: $\newline$ For $K = 70.5$ degrees: angle $L = 180$ degrees $- 74$ degrees $- 70.5$ degrees $\approx 35.5$ degrees $\newline$ For $K = 109.5$ degrees: angle $L = 180$ degrees $- 74$ degrees $- 109.5$ degrees $K = 70.5$ $0$ degrees
Solve for angle $L$ : The second case gives us a negative value for angle $L$ , which is not possible in a triangle. Therefore, the only valid value for angle $K$ is $70.5$ degrees.