Verified Answer :

calculista

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

\(\sqrt{3}\)

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

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