In a right triangle XYZ where angle X is 90 degrees, we can use the information provided to find the cosine of angle Z.
Given that tan Z = 3, we know that:
tanZ=oppositeadjacent=YZXZ=3.
Since triangle XYZ is a right triangle, we can use the Pythagorean theorem:
XY2+YZ2=XZ2.
Since angle X is 90 degrees, we have:
XZ2=XY2+YZ2=YZ2+YZ2=2YZ2.
Solving for YZ:
YZ2=XZ22.
Since we know the value of YZ/XZ (which is the tangent of angle Z):
tanZ=YZXZ=3.
Squaring both sides:
YZ2=9XZ2.
Now, substituting the value of YZ^2 from the Pythagorean theorem:
9XZ2=XZ22.
Solving for XZ:
XZ22=9XY2.
cosZ=12.