Can we say that a line is a particular case of a triangle

  • #1

Main Question or Discussion Point

I think we can say that a straight line is a triangle in which sum of two side equals third side in which two angle equals 0° and the middle angle equals 180°, but can we say that a striaght line is a triangle with two angle equals 90°, I am saying this because my sister is not beliving that sin(90°) = 1, she is saying that for proving sin(90°) = 1, I am making the triangle as a straight line but these ratios are only applicable for triangles , I told her that there will be no triangle with two angle equals 90° therefore I have to make it as a straight line.
 

Answers and Replies

  • #2
Simon Bridge
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The sine function is defined properly on a unit radius circle not a triangle.
When you do it this way, ##\theta## is the half-angle subtended by a chord. Half the length of the chord is the sine of the half-angle.
Draw a line of symmetry through the diagram to recover the usual soh cah toa triangles.

When the angle is 90deg, then the chord is just the diameter ... which is 2 for the unit circle.
Half the diameter is 1 ... hence sin(90)=1.

Similarly, if you draw a tangent to the circle at the symmetry line, the length of the tangent inside ##\theta## is called "the tangent of theta".
The distance along an angle line to the tangent is called the secant.

cosine, cotangent, and cosecant, are what yuo get with the above definitions using the complimentary angles.

Also see:
https://www.mathsisfun.com/geometry/unit-circle.html

You can argue that a rt-triangle becomes a line segment when one angle is zero. The other two angles are 90deg, two sides overlap and one has length zero... but it is easier to show on a circle.
 
  • #3
Wow great sir thank a lot

The sine function is defined properly on a unit radius circle not a triangle.
When you do it this way, ##\theta## is the half-angle subtended by a chord. Half the length of the chord is the sine of the half-angle.
Draw a line of symmetry through the diagram to recover the usual soh cah toa triangles.

When the angle is 90deg, then the chord is just the diameter ... which is 2 for the unit circle.
Half the diameter is 1 ... hence sin(90)=1.

Similarly, if you draw a tangent to the circle at the symmetry line, the length of the tangent inside ##\theta## is called "the tangent of theta".
The distance along an angle line to the tangent is called the secant.

cosine, cotangent, and cosecant, are what yuo get with the above definitions using the complimentary angles.

Also see:
https://www.mathsisfun.com/geometry/unit-circle.html

You can argue that a rt-triangle becomes a line segment when one angle is zero. The other two angles are 90deg, two sides overlap and one has length zero... but it is easier to show on a circle.
 
  • #4
Basicaly at the end u are saying that we can say that a right angle triangle will become straight line when one of its angle is zero. I got the idea of unit circle.thanks a lot sir.
 
  • #5
Simon Bridge
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It also works, and is less abstract, as a projection:
... if you have a stick upright on flat ground, with parallel light coming vertically down, then it's shadow has zero length.
If you tilt the stick an angle A to the vertical, the shadow gets longer ... when A=90deg, the shadow is the same length as the stick.
If the stick is 1 meter long, then the length of the shadow is sin(A) meters.
 

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