Final answer:
To solve for ∠ U in △ UVW with given sides and an angle, apply the Law of Sines to relate the sides to the angles, then use inverse trigonometric functions to find possible angle values, and ensure that the sum of the angles equals 180 degrees.
Step-by-step explanation:
To find all possible values of ∠ U to the nearest 10th of a degree in △ UVW, given u=7cm, w=3.7cm and ∠ W=19°, we can use the Law of Sines which relates the sides of a triangle to its angles.
According to the Law of Sines:
¬(Sin(∠ U) / u) = (Sin(∠ W) / w)
First, we calculate the ratio for ∠ W:
Sin(∠ W) / w = Sin(19°) / 3.7cm
Now, we can write the equation for ∠ U:
Sin(∠ U) / 7cm = Sin(19°) / 3.7cm
This will yield the sine of angle U:
∠ U = arcsin((Sin(19°) / 3.7cm) × 7cm)
Calculating this will give us the value of ∠ U. Since the sine function has a range of [-1, 1], meaningful results for the arcsine function (which will give us the angle in degrees) will be obtained only if the calculated sine is within this range. If multiple values exist because of the sine function's periodic nature, we take the one that results in a valid triangle (where the sum of the angles is 180 degrees).
Lastly, we must remember that in any triangle, the sum of the angles must equal 180 degrees. If we find one value for ∠ U, we can check to see if there is another possible value by subtracting the first value from 180° and subtracting ∠ W as well to see if a second valid triangle is possible.
Using the steps above, we can find the possible value or values of ∠ U, rounded to the nearest tenth of a degree, completing this trigonometry problem.