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In /_\MNO,n=57 inches, m=69 inches and /_M=58^(@). Find all possible values of /_N, to the nearest 10th of a degree. Answer:

In △MNO,n=57 \triangle \mathrm{MNO}, n=57 inches, m=69 m=69 inches and ∠M=58∘ \angle \mathrm{M}=58^{\circ} . Find all possible values of ∠N \angle \mathrm{N} , to the nearest 1010th of a degree.\newlineAnswer:

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Q. In △MNO,n=57 \triangle \mathrm{MNO}, n=57 inches, m=69 m=69 inches and ∠M=58∘ \angle \mathrm{M}=58^{\circ} . Find all possible values of ∠N \angle \mathrm{N} , to the nearest 1010th of a degree.\newlineAnswer:
  1. Apply Law of Cosines: To find the angle ∠N\angle N, 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 Law of Cosines states that for any triangle ABCABC with sides aa, bb, and cc opposite angles AA, BB, and CC respectively, c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab\cos(C). In our case, we can let side mm be cc, side ABCABC11 be bb, and angle ABCABC33 be angle CC.
  2. Convert angle to radians: First, we need to convert angle ∠M\angle M from degrees to radians because the cosine function in most calculators requires the angle to be in radians. However, since we are looking for an angle and not calculating a cosine value directly, we can skip this step and use the degree measure directly in the Law of Cosines.
  3. Use Law of Cosines to find cos⁡(N)\cos(N): Now, we apply the Law of Cosines to solve for cos⁡(∠N)\cos(\angle N):n2=m2+o2−2⋅m⋅o⋅cos⁡(∠N)n^2 = m^2 + o^2 - 2 \cdot m \cdot o \cdot \cos(\angle N),where oo is the length of the side opposite to angle ∠N\angle N, which we don't know yet, and nn and mm are the known sides. However, we realize that we cannot use the Law of Cosines directly to find angle ∠N\angle N because we do not have the length of side oo.

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