Why does the tangent function behave like the angle itself on a unit circle?

In summary, the discussion is about the intuitive and geometrical perspective of allowing tan alpha to be equal to alpha for small values of alpha. The reasoning behind this is that when comparing the length along the tangent of a circle with radius R inside an angle alpha to the arclength of the same circle, the difference becomes negligible when R is much larger than s. This concept is further explained using the definitions of tangent, sine, and cosine in relation to a unit circle.
  • #1
member 428835
hey all

can anyone explain why, for small [itex] \alpha[/itex] we may allow [itex] \tan \alpha = \alpha[/itex] at an intuitive, geometrical perspective. i already understand the series explanation and higher order of tangent. I am just trying for a picture.

thanks!
 
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  • #2
Because it is a very good approximation.

To see why: what is the slope of the tangent at ##\alpha=0##?More exactly - look at the definition of a tangent:

The length along the tangent to a circle radius R inside the some angle ##\alpha## is ##t=R\tan\alpha##
The arclength of a circle inside the same angle ##\alpha## is ##s=R\alpha##

When R>>s, then someone standing on the surface thinks the circle is actually flat.
i.e. it looks to be the same distance as the flat tangent measure. So ##t\approx s##
 
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  • #3
PF for the win! thanks simon
 
  • #4
Cool!

By the same token:
sinA = A
cosA = 1

When you realize that the trig functions are the names of lengths defined on a unit circle the whole thing makes a lot more sense.
 

What is the tangent function?

The tangent function, also known as the tangent ratio, is a mathematical function that relates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. It is commonly used in trigonometry and has applications in various fields such as physics and engineering.

How is the tangent function calculated?

To calculate the tangent of an angle, divide the length of the opposite side by the length of the adjacent side. This can be done using a scientific calculator or by using trigonometric tables. The result is a decimal number that represents the tangent ratio of the angle.

What is the range and domain of the tangent function?

The range of the tangent function is all real numbers, while the domain is restricted to angles between -π/2 and π/2 radians or -90° and 90° degrees. This is because the tangent function is undefined at angles of 90° and -90°, as the length of the adjacent side is equal to 0.

What are the properties of the tangent function?

The tangent function has several properties, including being periodic with a period of π radians or 180 degrees, having a vertical asymptote at 90° and -90°, and having an odd symmetry. It also has an inverse function, the arctangent, which allows for solving for angles given the tangent ratio.

How is the tangent function used in real life?

The tangent function has many practical applications in fields such as geometry, navigation, and physics. It is used to calculate distances and angles in real-world situations, such as determining the height of a building or the slope of a road. It is also used in computer graphics to create realistic 3D images and in engineering to design structures and machines.

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