In triangle ΔXYZ, XY = 15, YZ = 21, and XZ = 27. What is the measure of angle Z to the nearest degree?
The bot got it wrong!
the side opposite angle Z is 15, not 21, so the equation should be:
15^2 = 27^2 + 21^2 - 2(27)(21)cosZ
1135cosZ = 945
cosZ = 945/1135 = 5/6
angle Z = appr 33.56°
To find the measure of angle Z, we can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the side opposite to angle C, and a and b are the other two sides.
In this case, we have side XY as a = 15, side YZ as b = 21, and side XZ as c = 27.
Substituting the values into the formula, we get:
27^2 = 15^2 + 21^2 - 2 * 15 * 21 * cos(Z)
729 = 225 + 441 - 630 * cos(Z)
729 = 666 - 630 * cos(Z)
Now, solve for cos(Z):
630 * cos(Z) = 666 - 729
630 * cos(Z) = -63
cos(Z) = -63 / 630
cos(Z) = -1/10
To find the measure of angle Z, we can take the inverse cosine (cos^(-1)) of -1/10 using a calculator.
cos^(-1)(-1/10) ≈ 101.54 degrees
Therefore, the measure of angle Z to the nearest degree is 102 degrees.
To find the measure of angle Z in triangle XYZ, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we have side XY = 15, side YZ = 21, and side XZ = 27. We are looking for angle Z.
Let's substitute the values into the Law of Cosines equation:
27^2 = 15^2 + 21^2 - 2 * 15 * 21 * cos(Z)
Now, we can solve for cos(Z):
729 = 225 + 441 - 630 * cos(Z)
729 = 666 - 630 * cos(Z)
630 * cos(Z) = 666 - 729
630 * cos(Z) = -63
cos(Z) = -63 / 630
cos(Z) = -0.1
Now, we can calculate the value of angle Z using the inverse cosine function (cos^(-1)):
Z = cos^(-1)(-0.1)
Using a calculator, the inverse cosine of -0.1 is approximately 105.2 degrees.
Therefore, the measure of angle Z to the nearest degree is approximately 105 degrees.
To find the measure of angle Z, we can use the Law of Cosines which states that:
c^2 = a^2 + b^2 - 2ab*cos(C)
Where c is the side opposite to angle C, and a and b are the other two sides.
In this case, we can choose side XY to be a, side XZ to be b, and side YZ to be c. Then we have:
YZ^2 = XY^2 + XZ^2 - 2*XY*XZ*cos(Z)
Substituting the given values:
21^2 = 15^2 + 27^2 - 2*15*27*cos(Z)
Simplifying:
441 = 144 + 729 - 810*cos(Z)
Solving for cos(Z):
cos(Z) = (144 + 729 - 441) / (2*15*27)
cos(Z) = 0.5741
Taking the inverse cosine of both sides:
Z = cos^-1(0.5741)
Z ≈ 55.6 degrees
Therefore, the measure of angle Z to the nearest degree is 56 degrees.