Answer:
- sin(X) = 6/7.5
- XY = 4.5
- cos(X) = 4.5/7.5
- tan(X) = 6/4.5
Step-by-step explanation:
It is convenient to use the Pythagorean theorem to find XY to start with. That theorem tells you ...
XZ² = YZ² + XY²
Solving for XY, you find ...
XY² = XZ² - YZ²
XY = √(XZ² - YZ²) = √(7.5² -6²) = √(56.25 -36) = √20.25
XY = 4.5
The mnemonic SOH CAH TOA is very helpful here. It reminds you that ...
Sin = Opposite/Hypotenuse
sin(X) = 6/7.5
Cos = Adjacent/Hypotenuse
cos(X) = 4.5/7.5
Tan = Opposite/Adjacent
tan(X) = 6/4.5
_____
Comment on the triangle and ratios
The side lengths of this triangle are in the ratios ...
XY : YZ : XZ = 3 : 4 : 5
If you recognize that the given sides are in the ratio 4 : 5, this tells you that you have a "3-4-5" right triangle with a scale factor of 1.5. At least, you can find XY = 1.5·3 = 4.5 with no further trouble.
The trig ratios could be reduced to sin(X) = 4/5; cos(X) = 3/5; tan(X) = 4/3, but the wording "don't simplify" suggests you want the numbers shown on the diagram, not their reduced ratios.
