Understanding Trigonometric Functions and Their Geometric Meaning

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Trigonometric functions like sine, cosine, and tangent extend beyond right triangles and are defined using the unit circle, where angles can be obtuse. The cosine and sine of an angle correspond to the x and y coordinates of a point on the unit circle, respectively. This allows for the calculation of trigonometric values for angles greater than 90 degrees, even though right triangles cannot contain obtuse angles. The negative values of these functions arise from the coordinates of points in certain quadrants of the unit circle, where either the x or y coordinate is negative. Understanding these concepts clarifies the geometric meaning of trigonometric functions across all angles.
ViolentCorpse
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I'm not sure if this is the correct section for this thread since this isn't homework, but my question is very basic, so I think this section is suitable.

I have two questions regarding the trigonometric functions (sinx,cosx,tanx etc).

1) What is the geometric meaning (i.e in the context of a right-angle triangle) of functions like cos(180°) or any of the trigonometric functions with an obtuse angle? I mean a right-angle triangle can not contain an obtuse angle so I don't understand how we can have ratios of two sides of a right-triangle that isn't even a right-triangle to begin with (if that makes any sense)?

2) Why are these functions negative for some angles? Since these are just ratios of any two lengths of a right-triangle and lengths usually aren't negative.

Pardon my ignorance and thank you for your precious time! :)
 
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ViolentCorpse said:
I'm not sure if this is the correct section for this thread since this isn't homework, but my question is very basic, so I think this section is suitable.

I have two questions regarding the trigonometric functions (sinx,cosx,tanx etc).

1) What is the geometric meaning (i.e in the context of a right-angle triangle) of functions like cos(180°) or any of the trigonometric functions with an obtuse angle? I mean a right-angle triangle can not contain an obtuse angle so I don't understand how we can have ratios of two sides of a right-triangle that isn't even a right-triangle to begin with (if that makes any sense)?

2) Why are these functions negative for some angles? Since these are just ratios of any two lengths of a right-triangle and lengths usually aren't negative.

Pardon my ignorance and thank you for your precious time! :)
The trig functions are not limited to angles in right triangles. They are defined in terms of the coordinates of points on the unit circle. The angle is the one between two rays: the ray from (0, 0) to (1, 0), and another ray from the origin to an arbitrary point on the unit circle. The x-coordinate of this point is the cosine of the angle, and the y-coordinate of this point is the sine of the angle. See http://en.wikipedia.org/wiki/Unit_circle.
 
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Oh. That was very helpful. Thanks a lot, Mark44! :)
 

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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