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In the above triangle, if x = , y = 1, and z = 2, then which of the following is equal to tan(60°)?
A. [tex]\sqrt{3}[/tex]
B. [tex]\frac{\sqrt{2} }{2}[/tex]
C. [tex]\frac{\sqrt{3} }{2}[/tex]
D. [tex]\frac{\sqrt{3} }{3}[/tex]

In the above triangle if x y 1 and z 2 then which of the following is equal to tan60 A texsqrt3tex B texfracsqrt2 2tex C texfracsqrt3 2tex D texfracsqrt3 3tex class=

Respuesta :

luisejr77

Answer: Option A

[tex]tan(60\°) = \sqrt{3}[/tex]

Step-by-step explanation:

We know the sides and z. So since it is a straight triangle we use the Pythagorean theorem to pull the length of the x side.

[tex]z ^ 2 = x ^ 2 + y ^ 2\\\\x^2 = z^2 - y^2\\\\x=\sqrt{z^2 - y^2}\\\\x=\sqrt{2^2 - 1^2}\\\\x=\sqrt{4 - 1}\\\\x=\sqrt{3}[/tex]

By definition, the tangent of an angle is:

[tex]tan(\theta) = \frac{opposite}{adjacent}[/tex]

In this case:

[tex]adjacent = y=1\\\\opposite=x =\sqrt{3}\\\\\theta=60\°[/tex]

Then:

[tex]tan(60\°) = \frac{\sqrt{3}}{1}[/tex]

[tex]tan(60\°) = \sqrt{3}[/tex]

mixter17

Hello!

The answer is:

The correct option is:

A. [tex]\sqrt{3}[/tex]

Why?

Since we already know the hypothenuse and the opposite side of the triangle (y), we can calculate the value of "x" using the Pythagorean Theorem.

We have that:

[tex]Hypothenuse^{2}=Adjacent^{2}+Opposite^{2}[/tex]

We know that:

[tex]Hypothenuse=z=2\\Adjacent=x=1[/tex]

So, substituting and calculating we have:

[tex]2^{2}=1^{2}+Opposite^{2}[/tex]

[tex]4-1=Opposite^{2}[/tex]

[tex]Opposite^{2}=3\\Opposite=\sqrt{3}[/tex]

Then,using the following trigonometric relation:

[tex]Tan(\alpha)=\frac{Opposite}{Adjacent}\\\\Tan(60\°)=Tan(\frac{Opposite}{Adjacent})=Tan(\frac{\sqrt{3} }{1})^=\sqrt{3[/tex]

We have that the correct option is:

A. [tex]\sqrt{3}[/tex]

Have a nice day!

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