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Thread: Definition of sine or cosine

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    Definition of sine or cosine

    Sine and cosine are often introduced at early school mathematics as ratios of sides of triangles. Later ( certainly in the UK) we look at sine and cosine as coordinates on a unit circle. What is the correct definition of sine and cosine? Is sine defined as the y coordinate of a point on the unit circle? can we use this as the starting point?
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    Re: Definition of sine or cosine

    The ratios of triangle sides and points on the unit circle are equivalent. You can always fit a right triangle between, the origin, the point, and the points projection onto the x-axis.

    A more interesting possible definition of sine and cosine is their MacLaurin series.

    I wouldn't say there is a "correct" definition, but several equivalent definitions.

    However if pressed I'd go with the ratio of right triangle sides as this is generally the first one learned and was probably the first historical definition.
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    Re: Definition of sine or cosine

    Quote Originally Posted by romsek View Post
    The ratios of triangle sides and points on the unit circle are equivalent. You can always fit a right triangle between, the origin, the point, and the points projection onto the x-axis.

    A more interesting possible definition of sine and cosine is their MacLaurin series.

    I wouldn't say there is a "correct" definition, but several equivalent definitions.

    However if pressed I'd go with the ratio of right triangle sides as this is generally the first one learned and was probably the first historical definition.

    Is it the case that the ratio of sides is only applicable for angles less than 90 degrees and the points on the unit circle extend this?
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    Re: Definition of sine or cosine

    Quote Originally Posted by rodders View Post
    Is it the case that the ratio of sides is only applicable for angles less than 90 degrees and the points on the unit circle extend this?
    What the unit circle does is account for the trig values being signed. The sides ratios will only ever provide a positive value.

    This is equivalent to what you mention regarding the domain.

    Personally I like the series definitions myself. They let the functions to remain unassociated with any geometry. When we use sinusoids as signals we certainly don't care about any geometric origins.

    At any rate it doesn't really matter. The definitions are all equivalent.
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    Re: Definition of sine or cosine

    Quote Originally Posted by rodders View Post
    Is it the case that the ratio of sides is only applicable for angles less than 90 degrees and the points on the unit circle extend this?
    Sine, Cosine and Tangent apply only to right triangles. The sum of the angles in a triangle is 180 degrees. By definition, in a right triangle, one of the angles must be 90 degrees. Therefore each of the other two angles must be less than 90 degrees. Consequently, the ratio of sides (Sine, Cosine and Tangent) is only applicable for angles less than 90 degrees. Sine, Cosine and Tangent do not apply to the 90 degree angle!

    Years ago my 10th grade teacher gave our class a phrase tp help us remember how to calculate Sine, Cosine and Tangent. His phrase was "SOH - CAH - TOA". Pronounced "So -Ca - Toe - a". S, C and T are Sine, Cosine and Tangent. SOH prompts Sine is calculated as Opposite side divided by Hypotenuse. CAH prompts Cosine is calculated as Adjacent side divided by Hypotenuse. TOA prompts Tangent is calculated as Opposite side divided by Adjacent side. I've remembered "SOH - CAH - TOA" all the years since 10th grade and have applied it innumerable times when I needed to calculate Sine, Cosine or Tangent.

    Steve
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    Re: Definition of sine or cosine

    Hey rodders.

    The trigonometric functions make geometry consistent in both the real and the complex numbers.

    Geometry studies distance and angle and tries to figure out how to organize information with these two properties.

    The sines and cosines typically deal with what is called negative curvature meaning that a geometric object loops back on itself - like a circle.

    Other kinds of geometry look at flat objects [like a plane] and positively curved objects [like a saddle that you put on a horse].

    Everything that uses these functions of sines, cosines, tangents, and other derivatives makes all of geometry with negative curvature consistent and they work with all combinations of variables that are real numbers or complex numbers.

    When you try and find what works, you end up getting the trigonometric functions. A lot of mathematics is done the same way in that you find [or the mathematicians do anyway] what works based on trying to get something that doesn't break down when you try and use it in all the situations.
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