Why is this true about the angle theta

In summary, the conversation discusses the small angle approximation of the sine function and how it relates to the equation V_{e} \times sin \Theta \cong V_{e}\Theta. The answer is explained using the Taylor Expansion for sine and the assumption that \Theta \rightarrow 0. This approximation can be written as \sin{\theta} \approx \theta, making the equation clear.
  • #1
Pietair
59
0

Homework Statement


From my notes:

Picture2.png


Why is:
[tex]V_{e} \times sin \Theta \cong V_{e}\Theta[/tex] ?

The only thing I know is that the assumption: [tex]\Theta \rightarrow 0 [/tex] has been made, but that doesn't make the above equation clear to me...

Thanks in advance!
 
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  • #2


I'm not 100% sure what you're asking, but it looks like you are just talking about the small angle approximation of the sine function. That's easy to show if you know the Taylor Expansion for sine.

[tex]\sin{\theta} = \theta - {\theta^3 \over 3!} + {\theta^5 \over 5!} - ...[/tex]

Now, since [tex]\theta[/tex] is very small, any positive integer power of [tex]\theta[/tex] will go to 0 even faster than [tex]\theta[/tex] itself. ie. [tex]\theta^3 \to 0[/tex], [tex]\theta^5 \to 0[/tex], etc.

So [tex]\sin{\theta} \approx \theta[/tex]
 
  • #3


That was exactly my question, the answer is clear, thanks a lot!
 

1. Why is the angle theta measured in radians instead of degrees?

The use of radians in measuring angles is more convenient and efficient in mathematical calculations compared to degrees. Radians are based on the concept of the radius of a circle, making it easier to relate to the geometry of circles and curved objects. It also simplifies many trigonometric functions and their derivatives, making it a preferred unit of measurement in mathematics and science.

2. Why does the value of theta range from 0 to 360 degrees or 0 to 2π radians?

The range of values for theta is based on the concept of a complete rotation in a circle. A full rotation is defined as 360 degrees or 2π radians. This allows us to measure any angle within this range and easily convert between degrees and radians.

3. Why is theta often used in trigonometric functions?

Trigonometric functions, such as sine, cosine, and tangent, are used to relate the sides of a right triangle to its angles. Since theta represents an angle in a triangle, it is commonly used in these functions to solve for unknown sides or angles. It is also used in other mathematical and scientific applications, such as in polar coordinates and complex numbers.

4. Why is theta used as a variable in equations and formulas?

Theta is a common variable used in mathematics and science to represent an unknown angle. It allows for a general representation that can be applied to various scenarios and problems. It is also often used in conjunction with other variables, such as radius or height, to describe the relationship between different quantities in a problem.

5. Why does theta have different meanings in different fields of study?

The use of theta as a variable and its representation of an angle is not limited to mathematics. It is also used in fields such as physics, engineering, and astronomy to represent different quantities and concepts. Its meaning and interpretation may vary depending on the context in which it is used, highlighting its versatility and importance in various areas of study.

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