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In /_\UVW,w=8.9 inches, v=9 inches and /_V=50^(@). Find all possible values of /_W, to the nearest 10th of a degree. Answer:

In △UVW,w=8.9 \triangle \mathrm{UVW}, w=8.9 inches, v=9 v=9 inches and ∠V=50∘ \angle \mathrm{V}=50^{\circ} . Find all possible values of ∠W \angle \mathrm{W} , to the nearest 1010th of a degree.\newlineAnswer:

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Q. In △UVW,w=8.9 \triangle \mathrm{UVW}, w=8.9 inches, v=9 v=9 inches and ∠V=50∘ \angle \mathrm{V}=50^{\circ} . Find all possible values of ∠W \angle \mathrm{W} , to the nearest 1010th of a degree.\newlineAnswer:
  1. Use Law of Sines: To find the possible values of ∠W\angle W, 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 opposite angle is constant for all sides and angles in the triangle. The formula is:\newlineasin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\newlineHere, we have side vv opposite ∠V\angle V, and we want to find ∠W\angle W opposite side ww.\newlineFirst, we need to find the sine of ∠V\angle V.\newlinesin⁡(∠V)=sin⁡(50∘)\sin(\angle V) = \sin(50^\circ)
  2. Calculate sine of 5050 degrees: Now we calculate the sine of 5050 degrees using a calculator. \newlinesin⁡(50∘)≈0.7660\sin(50^\circ) \approx 0.7660
  3. Set up ratios for sides: Next, we set up the ratio for side vv and ∠V\angle V, and side ww and ∠W\angle W using the Law of Sines.\newlinevsin⁡(∠V)=wsin⁡(∠W)\frac{v}{\sin(\angle V)} = \frac{w}{\sin(\angle W)}\newline9sin⁡(50∘)=8.9sin⁡(∠W)\frac{9}{\sin(50^\circ)} = \frac{8.9}{\sin(\angle W)}
  4. Solve for sin⁡(∠W)\sin(\angle W): We substitute the value of sin⁡(50∘)\sin(50^\circ) into the equation.90.7660=8.9sin⁡(∠W)\frac{9}{0.7660} = \frac{8.9}{\sin(\angle W)}
  5. Calculate sin⁡(∠W)\sin(\angle W): Now we solve for sin⁡(∠W)\sin(\angle W).sin⁡(∠W)=8.9×0.76609\sin(\angle W) = \frac{8.9 \times 0.7660}{9}sin⁡(∠W)≈0.7660×0.9889\sin(\angle W) \approx 0.7660 \times 0.9889
  6. Check range of sin⁡(∠W)\sin(\angle W): We calculate the value of sin⁡(∠W)\sin(\angle W).sin⁡(∠W)≈0.7579\sin(\angle W) \approx 0.7579
  7. Find angle ∠W\angle W: Since the sine function has a range of −1-1 to 11, and we are dealing with a triangle, the value of sin⁡(∠W)\sin(\angle W) must be between 00 and 11. The value we obtained is within this range, so we can proceed to find the angle ∠W\angle W. We use the inverse sine function (arcsin⁡\arcsin) to find the angle whose sine is 0.75790.7579. ∠W≈arcsin⁡(0.7579)\angle W \approx \arcsin(0.7579)
  8. Consider ambiguous case: We calculate the angle using a calculator.\newline∠W≈arcsin⁡(0.7579)\angle W \approx \arcsin(0.7579)\newline∠W≈49.5\angle W \approx 49.5 degrees
  9. Calculate supplement of ∠W\angle W: However, since we are dealing with a triangle, there is a possibility of having two different angles with the same sine value, known as the ambiguous case of the Law of Sines. This occurs when the angle is acute (less than 9090 degrees). The other possible angle would be the supplement of the angle we found, which is 180180 degrees - ∠W\angle W. \newline∠W′=180\angle W' = 180 degrees −49.5- 49.5 degrees
  10. Check validity of angles: We calculate the supplement of ∠W\angle W. ∠W′=180∘−49.5∘\angle W' = 180^\circ - 49.5^\circ ∠W′≈130.5∘\angle W' \approx 130.5^\circ
  11. Valid value for /_W: Now we have two possible values for /_W: 49.549.5 degrees and 130.5130.5 degrees. However, we must check if both of these angles are valid in the context of a triangle. The sum of the angles in any triangle must be 180180 degrees. We already have one angle, /_V, which is 5050 degrees. Let's add /_V to the first possible value of /_W and see if the sum is less than 180180 degrees.\newline5050 degrees + 49.549.5 degrees = 99.599.5 degrees
  12. Valid value for /_W: Now we have two possible values for /_W: 49.549.5 degrees and 130.5130.5 degrees. However, we must check if both of these angles are valid in the context of a triangle. The sum of the angles in any triangle must be 180180 degrees. We already have one angle, /_V, which is 5050 degrees. Let's add /_V to the first possible value of /_W and see if the sum is less than 180180 degrees.\newline5050 degrees + 49.549.5 degrees = 99.599.5 degreesSince 99.599.5 degrees is less than 180180 degrees, there is room for a third angle, so 49.549.5 degrees is a valid value for /_W. Now let's check the second possible value.\newline5050 degrees + 130.5130.5 degrees = 130.5130.533 degrees
  13. Valid value for /_W: Now we have two possible values for /_W: 49.549.5 degrees and 130.5130.5 degrees. However, we must check if both of these angles are valid in the context of a triangle. The sum of the angles in any triangle must be 180180 degrees. We already have one angle, /_V, which is 5050 degrees. Let's add /_V to the first possible value of /_W and see if the sum is less than 180180 degrees. We find that the sum of /_V and the first possible value of /_W is 99.599.5 degrees, which is less than 180180 degrees, indicating that this value of /_W is potentially valid. However, when we add /_V to the second possible value of /_W, we get a sum of 180.5180.5 degrees, which exceeds the total possible sum of angles in a triangle, indicating that this value of /_W is not valid. Therefore, only the first value of /_W, 49.549.5 degrees, is considered valid in the context of a triangle. Regarding the angles of /_W that we are considering, it is important to note that the validity of these angles is determined based on the fundamental property that the sum of the angles in any triangle must not exceed 180180 degrees. This property is crucial in determining the validity of the potential values of /_W that we are examining.

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