SOLUTION: X Y Z is a right triangle in Y where XY = YZ = 5 cm. Find: 1. Cos (Z) 2. Tan (Z) 3. Csc (Z) 4. Sec (Z)

Question 1132824: X Y Z is a right triangle in Y where XY = YZ = 5 cm. Find:
1. Cos (Z)
2. Tan (Z)
3. Csc (Z)
4. Sec (Z)
Found 2 solutions by Alex.33, Theo:
Answer by Alex.33(110) (Show Source): You can put this solution on YOUR website!
since in a right triangle a^2+b^2=c^2, a or b cannot be equal to c(the hypotenuse).
If either of XY or YZ is the hypotenuse, it would violate the above conclusion. Therefore, XY or YZ cannot be the hypotenuse; XZ is the hypotenuse.
Therefore, Y=90 degrees; X=Z=45 degrees.
So

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
xyz is the right triangle.
the 90 degree angle is at y.
xy = yz so this is an isosceles right triangle.
the hypotenuse of this right triangle is xz which is equal to sqrt (25 + 25) = sqrt(50).

the trigonometric functions are:

cos(z) = yz/xz = 5/sqrt(50)
tan(z) = xy/yz = 5/5 = 1
csc(z) = 1/sin(z) = 1/(xy/xz) = 1/(5/sqrt(50)) = sqrt(50)/5.
sec(z) = 1/cos(z) = 1/(yz/xz) = 1/(5/sqrt(50) = sqrt(50)/5.

since this is an isosceles right triangle, angle z must be equal to 45 degrees and also angle x must be equal to 45 degrees as well.

you can use your calculator to confirm that:

cos(45) = .7071067812 and 5/sqrt(50) = the same.
likewise, tan(45) = 1
likewise csc(45) = 1 / sin(45) = 1.414213562 and sqrt(50)/5 = the same.
likewise sec(45) = 1 / COS(45) = 1.414213562 and sqrt(50)/5 = the same.

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