Tan 60 Degrees

Trigonometry is a branch of Mathematics that deals with Triangles. The word Trigonometry is composed of two Greek words Trigōnon (meaning Triangle) and Metron (meaning measure).  So in short, we can say that measuring a Triangle (specifically right-Angled Triangle) is Trigonometry. Trigonometry is the study between the relationships dealing with Angles, Heights and Lengths of Triangles and also the relationships between the different Circle parts and other Geometric figures. In the field of Astronomy, Engineering, Architectural Design, and Physics, Trigonometry applications are found. Trigonometric Identities are very useful and help to solve the problems better by studying the formulae below. There are a huge number of fields in which these Trigonometry Identities and Trigonometric Equations are used. 

Tan 60 Degrees Value

A Right-Angled Triangle is a closed figure having three sides, three Angles and three edges such that one of the three Angles of a Triangle is of 90 degree. 

To find the value of tan 60 degrees, let us take one of the Angles 60 degrees. 

• Hypotenuse: The longest side of all the three sides of a Right-Angled Triangle which is also opposite to the Right Angle is called Hypotenuse.

• Base or Adjacent: The side containing 60 degrees as well as 90 degrees is called Base.

• Perpendicular or Opposite: The side perpendicular to base not containing 60 degrees is called perpendicular or opposite side of a right-Angled Triangle.

\[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]  

 \[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

Here, = 60 degrees.

\[\tan \left( \theta  \right)\frac{{\sin \theta }}{{\cos \theta }}\]

 

Thus, 

 

\[\frac{{\sin \left( \theta  \right)}}{{\cos \left( \theta  \right)}} = \frac{{\frac{{opposite}}{{Hypotenuse}}}}{{\frac{{Adjacent}}{{Hypotenuse}}}} = \frac{{Opposite}}{{Adjacent}} = \tan \left( \theta  \right)\]

 

\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]

 

Therefore, the value of tan 60 = Opposite / Adjacent.

The Triumphant Triangle:

First up is the equilateral triangle, which has all angles measuring 60°. This triangle is like the starting point for our trigonometric journey.

Now, if we cut that equilateral triangle in half, we create two smaller triangles. One of these new triangles is super special - it's called the 30-60-90 triangle. Why? Because its angles are 30°, 60°, and 90°. This triangle is like the key to unlocking trigonometric secrets.

In the 30-60-90 triangle, there's a neat trick. The lengths of its sides are connected by simple ratios. So, let's focus on tan 60° and its special ratio.

For sine (sin), it's about the opposite side over the hypotenuse, and for tan 60°, it's 1/√3 because the opposite side is half the hypotenuse in an equilateral triangle.

For cosine (cos), it's the adjacent side over the hypotenuse, and for tan 60°, it's √3/2 because the adjacent side is half the hypotenuse multiplied by √3.

Lastly, for tangent (tan), it's the opposite side over the adjacent side, and for tan 60°, it's √3, which you get by dividing sin by cos. Cool, right? Triangles are like secret code breakers in trigonometry!

What is the value of tan 60 Degrees?                      

We can use Geometry to find the value of tan 60 degrees. Let us consider an Equilateral Triangle ABC such that all the interior Angles are 60 degrees.

Therefore,

\[\angle A = \angle B = \angle C = 60^\circ \]

A perpendicular AD is drawn from A to BC. In Triangle ADB we can say

\[\angle {\text{ }}ADB = 90^\circ \] and in  Triangle ADC, 

\[\angle ADC = 90^\circ .\]Thus,  ∠ ADB =  ∠ ADC.

We also know that

\[\angle {\text{ }}ABD = \angle ACD = {\text{ }}60^\circ \]That means, AD=AD

Now, the question arises what is the value of the Tangent of 60 degrees. So, we know that according to AAS Congruence,

\[\Delta {\text{ }}ABD{\text{ }} \cong {\text{ }}\Delta {\text{ }}ACD\]

From this, we can say

BD = DC

Let us take, AB = BC =2a

Then, 

BD= ½ (BC) =½ (2a) =a

By using Pythagoras theorem,

• AB² = AD²- BD²

• AD² = AB²- BD²

• AD² = (2a)²- a²

• AD² = 4a²- a²

• AD² = 3a²

Therefore, AD=a√3

Now in Triangle ADB,

tan 60 = AD/BD = a√3/a = √3

Therefore, the exact value of tan 60 degrees is √3. We can also derive the values of Tan 0°, 30°, 45°, 90°, 180°, 270° and 360° in the same way. 

The values of the important Functions of Sin can also be determined by the given method:

Simplifying in a tabular form:

The value of cosine Functions is opposite if sine Functions as in:

Summary table of the value of sin and cosine angles:

From here, we can find out the value of tan for each angle by:  

\[\tan \left( \theta  \right)\frac{{\sin \theta }}{{\cos \theta }}\]

Applications of Tan 60 Degrees

Knowing tan 60 degrees is just the beginning! Armed with this crucial value, we can solve a multitude of problems across various fields:

• Geometry: Calculating side lengths and angles in triangles, quadrilaterals, and other polygonal shapes involving 60-degree angles.

• Navigation: Determining ship bearing angles in marine navigation or designing efficient airplane routes based on trigonometry.

• Physics: Understanding projectile motion, wave propagation, and forces acting on structures with 60-degree angles.

• Engineering: Designing bridges, dams, and other structures requiring precise calculations of angles and forces for stability.

Mastering the Tricks

To truly understand how to deal with tan 60 degrees, let's explore some practical methods:

1. Unit circle shortcut: Picture a circle with a radius of 1, where we highlight the point corresponding to 60 degrees. The coordinates of this point give us sine and cosine values. These values can then be used to figure out tan 60 degrees using trigonometric tricks.

2. Special triangle cheat: Memorize the sine, cosine, and tangent values for 30, 45, and 60 degrees. This knowledge makes quick calculations a breeze.

3. Double and half-angle shortcuts: Make use of these formulas to connect the trigonometric values of angles like 30 and 60 degrees back to tan 60 degrees. This simplifies the process and avoids unnecessary complications.

Solved Examples:

Question: Calculate the value of tan 75 degrees.

Solution: The value of tan 75 degrees can be calculated in many ways. One of the ways is given below:

75 can also be seen as a sum of 30 and 45 (because the value of Tan 30 nd Tan 45 can easily be placed as per the table)

 tan75 = tan(30+45)

Also, 

tan(A+B) = (tanA + tanB)/(1 - tanATanB)

So, tan(30+45) = (tan30+tan45)/(1+tan30tan45)

Now, tan30 = 1/✓3

And, tan45 = 1

tan(30+45) = (1/✓3 + 1)/(1+ 1/✓3)

Therefore, tan75 = (1 + 1/✓3)²

So, tan75 = 1 + 1/3 + 2/✓3 

      tan75 = 4/3 + 2/✓3

Thus, tan75 = (4✓3 + 6)/3✓3

The value of ✓3 is 1.73205

So, tan75 = 12.9282/5.19615 = 2.4880344

So, the approximate value of tan75 is 2.4880344

Conclusion

The value of tan 60° isn't just a random number; it has a fascinating story. It highlights the beauty of mathematics, showing how basic shapes and ratios can open up a world of possibilities. When you grasp the concept of tan 60°, you not only acquire a valuable tool in your math skills but also develop a greater appreciation for the captivating connection between geometry and numbers.