Question

Given right triangle XYZ, what is the value of tan(60°)?

Answer 1

we know that

Step $1$

in the right triangle XYZ

Applying the Pythagorean Theorem

Find the value of ZY

$XY^{2}=XZ^{2} +ZY^{2} \\ ZY^{2}=XY^{2}-XZ^{2}\\ ZY^{2}=42^{2}-21^{2}\\ ZY^{2}=1,323\\ ZY=\sqrt{1,323}\ units$

$ZY=21\sqrt{3}\ units$

Step $2$

in the right triangle XYZ

Find tan$60$°

we know that

$tan\ 60=\frac{opposite\ side\ angle\ 60}{adjacent\ side\ angle\ 60}$

in this problem

opposite side angle $60$ is ZY

adjacent side angle $60$ is XZ

so

$tan\ 60=\frac{ZY}{XZ} \\ \\ tan\ 60=\frac{21\sqrt{3}}{21} \\ \\ tan\ 60=\sqrt{3}$

therefore

the answer is

$\sqrt{3}$

Answer 2

You can use Pythagoras theorem along with the definition of tangent ratio to find out the value of tan(60 degrees).

The value of tan(60°) is $\sqrt{3}$

What does Pythagoras theorem says?

In a right angled triangle(one angle of 90°), the slant line's length's square(slant line is also called hypotenuse of that triangle) is equal to sum of square of length of other sides.

Thus, for given triangle(since it is a right angled triangle as angle Z is right angled(90 degrees), we have:

$(XY)^2 = (XZ)^2 + (ZY)^2$

What is tangent ratio of angle in a right angled triangle?
$tan(\theta) = \dfrac{\text{Opposide side}}{\text{Adjacent base side}}$

Using both above definitions

We have:

$(XY)^2 = (XZ)^2 + (ZY)^2\\(YZ)^2 = (XY)^2 - (XZ)^2\\\\YZ = \sqrt{ (XY)^2 - (XZ)^2}\\ \\\text{\: \: (positive root since YZ is length and thus non negative})}$

Thus, we have:

$YZ = \sqrt{42^2 - 21^2} = \sqrt{1323} = 21\sqrt{3}$

And thus,

$\tan(60^\circ) = \dfrac{YZ}{XZ} = \dfrac{21\sqrt{3}}{21} = \sqrt{3}$

Thus,

The value of tan(60°) is $\sqrt{3}$

Learn more about trigonometric ratios here:

https://brainly.com/question/22599614