what is the definition of cosine , sine

Question

I know that sine is the ratio of the perpendicular to the hypotenuse of an acute angle. Similarly cosine is the ratio of the base and hypotenuse .

But now I found that there is sine and cosine of an angle over 90 degrees or negative degrees . I do not understand how they can determine the value of these sine and cosine . I have heard about the unit circle , and I find it is not convincing to determined the value of sine and cosine just by looking at the picture . I mean , is there any others method like using equations or theorems to calculate the cosine of an over 90 degrees?

John Joy · Answer 1 · 2015-07-27T18:43:43.520

Suppose that the black dot on the rear wheel (where the wheel connects with the coupling rod) was a distance of $1$ from the center of the wheel, and its initial position was directly to the right of the center of the wheel.

Given that the wheel had turned and angle of $\theta$ (from its initial position), Vertically, how far above the center of the wheel would the black dot be? Horizontally, how far to the right of the the center of the wheel would the black do be (a negative value would mean that it was to the left)?

Try drawing some diagrams of the wheel in different positions (drawing reference triangles would be helpful here).

Also try drawing a triangle with a hypotenuse of $1$. If $\theta$ is known, what would the length of the opposite side be? What would the length of the adjacent side be?

score 1 · Answer 2 · answered Jul 27 '15 at 06:37

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There are many definitions, but geometrically I believe the unit-circle definition is the best. It allows you to understand easly many trigonometric relations.

answered Jul 27 '15 at 06:37

Scientifica

8,781

score 1 · Answer 3 · answered Jul 27 '15 at 18:36

1

When you press the cos button on your calculator it probably takes use of the following definition:

cos(x) = $\sum_{0}^{\infty}\frac{(-1)^n x^{2n}}{2n!}$

answered Jul 27 '15 at 18:36

user256916

101
5

2

Depends on the calculator. Older calculators use the CORDIC algorithm. – Ben Grossmann Jul 27 '15 at 18:42
This is extremely unhelpful based on the level of knowledge of the question. – Aloizio Macedo Jul 27 '15 at 19:27
While this a possible definition of cosine, it is not very helpful intuitively for someone beginning off. I think it's better to derive this result assuming the unit curcle definition through Taylor Series. – Cataline Jul 27 '15 at 19:36

what is the definition of cosine , sine

3 Answers3