In \( \triangle N O P, n=62 \) inches, \( o=55 \) inches and \( p=35 \) inches. Find the measure of \( \angle O \) to the nearest 10 th of a degree.
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To find the measure of \( \angle O \) in triangle \( NOP \) using the Law of Cosines, we start with the formula: \[ o^2 = n^2 + p^2 - 2np \cdot \cos(O) \] Plugging in the values \( n = 62 \) inches, \( o = 55 \) inches, and \( p = 35 \) inches, we have: \[ 55^2 = 62^2 + 35^2 - 2 \cdot 62 \cdot 35 \cdot \cos(O) \] Calculating each term: \[ 3025 = 3844 + 1225 - 4340 \cdot \cos(O) \] Now, simplifying: \[ 3025 = 5069 - 4340 \cdot \cos(O) \] Rearranging gives: \[ 4340 \cdot \cos(O) = 5069 - 3025 \] \[ 4340 \cdot \cos(O) = 2044 \] Now, solving for \( \cos(O) \): \[ \cos(O) = \frac{2044}{4340} \approx 0.4710 \] Finally, finding \( O \): \[ O \approx \cos^{-1}(0.4710) \] Using a calculator: \[ O \approx 61.7^\circ \] Thus, the measure of \( \angle O \) to the nearest tenth of a degree is approximately \( 61.7^\circ \).
