Respuesta :

gmany

Answer:

[tex]\large\boxed{\sqrt3}[/tex]

Step-by-step explanation:

Look at the picture.

Use the Pythagorean theorem:

[tex]h^2+\left(\dfrac{a}{2}\right)^2=a^2[/tex]

solve for h:

[tex]h^2+\dfrac{a^2}{4}=a^2\qquad\text{subtract}\ \dfrac{a^2}{4}\ \text{from both sides}\\\\h^2=\dfrac{4a^2}{4}-\dfrac{a^2}{4}\\\\h^2=\dfrac{3a^2}{4}\to h=\sqrt{\dfrac{3a^2}{4}}\\\\h=\dfrac{\sqrt{3a^2}}{\sqrt4}\\\\h=\dfrac{a\sqrt3}{2}[/tex]

[tex]tangent=\dfrac{opposite}{adjacent}[/tex]

We have:

[tex]opposite=\dfrac{a\sqrt3}{2}\\\\adjacent=\dfrac{a}{2}[/tex]

Substitute:

[tex]\tan60^o=\dfrac{\frac{a\sqrt3}{2}}{\frac{a}{2}}=\dfrac{a\sqrt3}{2}\cdot\dfrac{2}{a}=\sqrt3[/tex]

Ver imagen gmany

Tangent (tan) is a trigonometric ratio, which holds values for each given angle tan(60°) can be expressed as [tex]\sqrt{3}[/tex].

Thus, second option  [tex]\sqrt{3}[/tex].

What are right angled triangles?

Right angled triangles are triangles containing one of their angle as right angle (90 degrees).

What is tangent?

Tangent (tan) in trigonometry is the ratio of perpendicular to base:[tex]tan(\theta) = \dfrac{\:\rm Perpendicular}{\:\rm Base}[/tex]

Perpendicular is the side opposite to the angle we're inputting in tan.

Base is the side which is perpendicular to that perpendicular side.

In the diagram of equilateral triangle given below, we have AB as Hypotenuse and BD as Base and AD as Perpendicular.

Since angle B  = 90 degrees and angle C = 60 degrees,thus angle A = 180 - 60 - 90 = 30 degrees.

Using Pythagoras theorem in triangle ABD

[tex]AB^2 = AD^2 + BD^2\\\\x^2 = AD^2 + (\dfrac{x}{2})^2\\\\AD = \sqrt{\dfrac{3x^2}{2^2}} = \dfrac{x}{2}\sqrt{3}\\[/tex]

Using definition of tangent we have

[tex]tan( \angle ABD) = \dfrac{AD}{BD}\\\\tan(60^\circ) = \dfrac{\sqrt{3}(\dfrac{x}{2})}{\dfrac{x}{2}}\\\\\\tan(60^\circ) = \sqrt{3}\\[/tex]

Thus, tan(60°) can be expressed as [tex]\sqrt{3}[/tex].

Learn more about trigonometric ratios here:

https://brainly.com/question/1201366

Ver imagen astha8579

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