Why Is Cosine Used to Calculate Angle Y in a Right Triangle?

[ai93]

Hi, this would be my first post of many in recent times as I have my maths exam soon! I am doing a lot of past papers and I need some help understand some questions.

In the triangle $$XYZ$$ angle $$Z$$ is a right angle. If $$XY$$= 15mm and $$YZ$$=8mm, calculate the angle $$Y$$, giving your answer in degrees accurate to 1dp.

The solution was;
$$COSY=\frac{8}{15}$$
$$\therefore Y=COS^{-1}\frac{8}{15}$$
=$$Y=57.8$$

Can someone clarify why COS was used, and how to attempt similar questions like this?

Continuing from this question,
Calculate the length of the side $$XZ$$

I see pythag theorem was used as the solution was
$$XZ^{2}=\sqrt{15^{2}-8^{2}}$$
$$XZ$$= $$\sqrt{161} = 12.7$$

but why?

 

[MarkFL]

In the given triangle, we know it is a right triangle, and we know the hypotenuse is:

$$\overline{XY}=15\text{ mm}$$

And we also know the side adjacent to $\angle Y$ is:

$$\overline{YZ}=8\text{ mm}$$

Now, since the cosine function is defined as:

$$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$

we may then state:

$$\cos(Y)=\frac{8\text{ mm}}{15\text{ mm}}=\frac{8}{15}$$

And so we find:

$$Y=\arccos\left(\frac{8}{15}\right)\approx57.8^{\circ}$$

edit: To answer the added part, by Pythagoras, we know the sum of the squares of the legs is equal to the square of the hypotenuse, which allows us to write:

$$\overline{XZ}^2+8^2=15^2$$

$$\overline{XZ}=\sqrt{15^2-8^2}=\sqrt{161}$$

 

[ai93]

In the given triangle, we know it is a right triangle, and we know the hypotenuse is:

$$\overline{XY}=15\text{ mm}$$

And we also know the side adjacent to $\angle Y$ is:

$$\overline{YZ}=8\text{ mm}$$

Now, since the cosine function is defined as:

$$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$

we may then state:

$$\cos(Y)=\frac{8\text{ mm}}{15\text{ mm}}=\frac{8}{15}$$

And so we find:

$$Y=\arccos\left(\frac{8}{15}\right)\approx57.8^{\circ}$$

edit: To answer the added part, by Pythagoras, we know the sum of the squares of the legs is equal to the square of the hypotenuse, which allows us to write:

$$\overline{XZ}^2+8^2=15^2$$

$$\overline{XZ}=\sqrt{15^2-8^2}=\sqrt{161}$$

Thank you. Much clearer now. Since the question involves a right angle, we must use the Pythag Theorem. SOHCAHTOA helps a lot in solving for Y too! :D