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In summary, the conversation discusses the concept of a straight line being a triangle with equal sum of two sides and two angles of 0°, as well as a triangle with two angles of 90°. The conversation also explains the proper definition of the sine function on a unit radius circle and the relationship between angles and chords. It is mentioned that a right triangle can be seen as a straight line when one angle is 0° and this concept is further illustrated using the example of a stick and its shadow.

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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.

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Simon Bridge said:

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.

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... 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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