In triangle XYZ,x=20.4,y=16.4,z=23.1,what is angle Z
Because you know all three sides you can use the Cosine Law. z^2=x^2 +y^2 -2xycosz and re-arrange to get cosz all by itself. Then solve for the angle z
To find angle Z in triangle XYZ, you can use the Law of Cosines. This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite angle C in triangle ABC.
In this case, we are looking for angle Z, so we can use the values of the sides x, y, and z:
x = 20.4
y = 16.4
z = 23.1
Now, we can substitute the values into the formula:
z^2 = x^2 + y^2 - 2xy * cos(Z)
Plugging in the values:
23.1^2 = 20.4^2 + 16.4^2 - 2(20.4)(16.4) * cos(Z)
Simplifying the equation:
533.61 = 416.16 + 268.96 - 670.08 * cos(Z)
Combining like terms:
533.61 = 685.12 - 670.08 * cos(Z)
Rearranging the equation:
670.08 * cos(Z) = 685.12 - 533.61
670.08 * cos(Z) = 151.51
Now, divide both sides by 670.08 to solve for cos(Z):
cos(Z) = 151.51 / 670.08
Using a calculator or math software, we can find the value of cos^-1(151.51 / 670.08):
cos^-1(0.226418) ≈ 77.02
Therefore, angle Z is approximately 77.02 degrees.
To find angle Z in triangle XYZ, we can use the Law of Cosines. The Law of Cosines states that in any 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 our case, we have side x = 20.4, side y = 16.4, and side z = 23.1. We want to find angle Z opposite side z.
Let's plug in the given values into the Law of Cosines equation:
z^2 = x^2 + y^2 - 2xy * cos(Z)
Substituting the values:
(23.1)^2 = (20.4)^2 + (16.4)^2 - 2 * 20.4 * 16.4 * cos(Z)
Simplifying the equation:
533.61 = 416.16 + 268.96 - 669.12 * cos(Z)
Combining like terms:
533.61 = 685.12 - 669.12 * cos(Z)
Rearranging the equation:
-669.12 * cos(Z) = 533.61 - 685.12
-669.12 * cos(Z) = -151.51
Dividing by -669.12:
cos(Z) = -151.51 / -669.12
cos(Z) ≈ 0.2265
To find angle Z, we need to calculate the inverse cosine (cos^-1) of 0.2265:
Z = cos^-1(0.2265)
Using a calculator or math tool, we find:
Z ≈ 77.54 degrees
Therefore, angle Z in triangle XYZ is approximately 77.54 degrees.