Answer:
Measure of ∠Y = 35°.
Explanation:
Because we don't know whether or not ∆XYZ is a right triangle, we'll need to use the law of cosines to find the measure of ∠Y to the nearest degree.
The Law of Cosines allows to solve triangles when all three sides or two sides and their included angle are given and has three equation depending on what we're solving for and what info we're given:
a^2 = b^2 + c^2 - 2bc * cos A
b^2 = a^2 + c^2 - 2ac * cos B
c^2 = a^2 + b^2 - 2ab * cos C
Thus, we can substitute a, b, c, and B and for x, y, z, and Y and plug in 7.8 for x, 5 for y, and 4.1 for z to find the measure of ∠Y:
Step 1: Plug in 7.8 for x, 5 for y, and 4.1 for z and simplify:
5^2 = 7.8^2 + 4.1^2 - 2(7.8)(4.1) * cos Y
25 = 60.84 + 16.81 - 63.96 * cos Y
25 = 77.65 - 63.96 * cos Y
Step 2: Subtract 77.65 from both sides:
(25 =77.65 - 63.96 * cos Y) - 77.65
-52.65 = -63.96 * cos Y
Step 3: Divide both sides by -63.96:
(-52.65 = -63.96 * cos Y) / -63.96
0.8231707317 = cos Y
Step 4; Take the inverse cosine of 0.8231707317 and round the answer to the nearest degree (aka the nearest whole number) to find the measure of angle Y:
cos^-1 (0.8231707317) = Y
34.59653258 = Y
35 = Y
Thus, the measure of ∠Y is about 35°.