Basic Trigonometry: Explaining the Rules

In summary, the definitions of sine, cosine, and tangent are derived from planar geometry. They can also be defined using the unit circle, in which case the proof for sine is immediate from similar triangles. However, there is still something to prove in showing that if two triangles have the same angle, then the quantity opposite side/hypotenuse is the same.
  • #1
iknownth
16
0
We all know that
sin theta = opposite side / hypotenuse
cos theta = adjacent side / hypotenuse
tan theta = opposite side / adjacent side.
But why? Are there some explanations behind or are they just defined by scientists?
 
Mathematics news on Phys.org
  • #2
iknownth said:
But why? Are there some explanations behind or are they just defined by scientists?

What do you mean by "explanations"?

To answer the question, they are the definitions used in planar geometry. There are other (equivalent) ways of defining them, but you'll get the same properties nonetheless.
 
  • #3
How can one prove that sin theta = opposite side / hypotenuse ?
 
  • #4
iknownth said:
How can one prove that sin theta = opposite side / hypotenuse ?

I'm assuming planar geometry here. There are two answers:
  1. It is the definition of sine. There is nothing to prove.
  2. The definition is using the unit circle. It which case the proof is immediate from similar triangles.

I guess it's worth asking: what is your definition of sine?
 
  • #5
pwsnafu said:
It is the definition of sine. There is nothing to prove.

In this case, there is still something to prove. You want to prove also that if two triangles have the same angle, then the quantity opposite side/hypothenuse is the same.
 
  • #6
micromass said:
In this case, there is still something to prove. You want to prove also that if two triangles have the same angle, then the quantity opposite side/hypothenuse is the same.

Arrgh yes of course.

Where's the brainfart smiley when you need one?
 

1. What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves using ratios and functions to solve for unknown values in a triangle.

2. What are the basic rules of trigonometry?

The basic rules of trigonometry include the sine, cosine, and tangent functions, which are used to find the ratios of the sides of a right triangle. These functions are often abbreviated as sin, cos, and tan, respectively.

3. How do I find the values of sine, cosine, and tangent?

To find the values of sine, cosine, and tangent, you can use a scientific calculator or refer to a trigonometric table. These values can also be found using the Pythagorean theorem and the definitions of the trigonometric functions.

4. What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is often used in trigonometry to find the lengths of sides in a triangle.

5. How is trigonometry used in real life?

Trigonometry has many practical applications in real life, such as in architecture, engineering, navigation, and astronomy. It can be used to calculate the height of buildings, the distance between two points, and the trajectory of a projectile, among other things.

Similar threads

  • General Math
Replies
1
Views
2K
  • General Math
Replies
6
Views
1K
  • General Math
Replies
1
Views
7K
Replies
4
Views
4K
  • Calculus and Beyond Homework Help
Replies
28
Views
1K
  • General Math
Replies
3
Views
905
  • Programming and Computer Science
Replies
16
Views
3K
  • General Math
Replies
2
Views
6K
  • General Math
Replies
8
Views
2K
Replies
7
Views
1K
Back
Top