Question

What are the values of cosY
and cosX
in the triangle below?

Right triangle X Y Z is shown with labeled sides. Side X Y is the hypotenuse and it is labeled the square root of 34. Side Y Z is the height and it is labeled 3. Side Z X is the base and it is labeled 5.

Answer 1

In right triangle XYZ, we have the following information:

- XY (the hypotenuse) = \(\sqrt{34}\)
- YZ (the height) = 3
- ZX (the base) = 5

### To find cosY:

The cosine of an angle in a right triangle is defined as the adjacent side divided by the hypotenuse. For angle Y, the adjacent side is ZX (5), and the hypotenuse is XY (\(\sqrt{34}\)).

\[
\cos Y = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{ZX}{XY} = \frac{5}{\sqrt{34}}
\]

### To find cosX:

For angle X, the adjacent side is YZ (3), and the hypotenuse is also XY (\(\sqrt{34}\)).

\[
\cos X = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{YZ}{XY} = \frac{3}{\sqrt{34}}
\]

### Final values:

- \(\cos Y = \frac{5}{\sqrt{34}}\)
- \(\cos X = \frac{3}{\sqrt{34}}\)