Find each measure using the given measures of triangleXYZ. Round angle measures to the nearest degree and side measures to the nearest tenth. Find measure of angle Y if y= 7,z=11,and the measure of angle Z=37.
The Law of Sines:
sinY/y = sinZ/z,
sinY/7 = sin37/11,
Multiply both sides by 7:
sinY = 7*sin37 / 11,
sinY = 0.3830,
Y = 22.5 DEG.
To find the measure of angle Y, we need to use the fact that the sum of the angles in a triangle is always 180 degrees.
Step 1: Find the measure of angle X.
Since we know the measure of angle Z is 37 degrees, and the sum of all three angles is 180 degrees, we can find angle X by subtracting the measures of angles Y and Z from 180 degrees.
180 - 37 = 143 degrees. So, angle X measures 143 degrees.
Step 2: Use the Law of Sines to find the length of side x.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Using this law, we can set up the following proportion:
x / sin(X) = 7 / sin(Y)
Now we can substitute the known values:
x / sin(143) = 7 / sin(Y)
Step 3: Solve for sin(Y).
To solve for sin(Y), we rearrange the equation above:
sin(Y) = (7 * sin(143)) / x
Step 4: Substitute the known values and solve for sin(Y).
sin(Y) = (7 * sin(143)) / x
sin(Y) = (7 * sin(143)) / 11
Using a calculator, we can calculate that sin(143) ≈ 0.6147.
sin(Y) = (7 * 0.6147) / 11 ≈ 0.3893
Step 5: Find the measure of angle Y.
To find the measure of angle Y, we need to use the inverse sine function (sin^(-1)).
Y = sin^(-1)(0.3893)
Using a calculator, we can find Y ≈ 22.6 degrees.
Therefore, the measure of angle Y is approximately 22.6 degrees.