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

  • Thread starter Thread starter ai93
  • Start date Start date
  • Tags Tags
    Cos Triangle
AI Thread Summary
Cosine is used to calculate angle Y in a right triangle because it relates the length of the adjacent side to the hypotenuse, expressed as cos(Y) = adjacent/hypotenuse. In triangle XYZ, with XY as the hypotenuse (15 mm) and YZ as the adjacent side (8 mm), the calculation yields Y = arccos(8/15), resulting in approximately 57.8 degrees. The Pythagorean theorem is then applied to find the length of side XZ, confirming that XZ^2 + YZ^2 = XY^2. This reinforces the understanding that in right triangles, both cosine and the Pythagorean theorem are essential for solving for angles and side lengths. The discussion highlights the importance of SOHCAHTOA in trigonometric calculations.
ai93
Messages
54
Reaction score
0
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?
 
Mathematics news on Phys.org
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:

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

$$\overline{XZ}=\sqrt{15^2-8^2}=\sqrt{161}$$
 
MarkFL said:
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
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top