Final answer:
To find the measure of angle w in triangle wxy, where the lengths of two sides and one angle are known, the Law of Sines is used. The student will substitute the given values into the Law of Sines formula and solve for angle w by using the inverse sin function, making sure to consider whether the angle is acute or obtuse.
Step-by-step explanation:
The student's question is about finding the measure of angle w in triangle wxy, where the lengths of two sides and the measure of one angle are provided. To find angle w, we can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles. This is expressed as a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the sides of the triangle, and A, B, and C are the angles opposite those sides, respectively.
Using the Law of Sines, we have:
Rearranging to solve for sin(w), we get:
sin(w) = (sin(x) × y) / w
Substituting the given values:
sin(w) = (sin(31°) × 310 cm) / 440 cm
Now, calculate the value of sin(w) and then use the inverse sin function to find angle w. The result should be rounded to the nearest degree.
It is important to remember that the sine function can have two different angles between 0° and 180° with the same sine value (since sine is positive in the first and second quadrants). Since we are dealing with a triangle, the angle sought must be less than 180°. If the calculated angle is obtuse, it can be directly accepted, but if the angle is acute, we must consider the possibility of the supplement as well. For an acute angle, we would subtract it from 180° to check for the obtuse possibility and then determine which angle is valid based on the context of the triangle (i.e., whether an obtuse angle is possible given the other known angle).
By solving this equation, we will find the measure of angle w in the triangle.