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We know that values of $\sin 400^{\circ} , \cos 135^{\circ} , \cos 0^{\circ}$ are $0.64, -0.71, 1$ respectively. But I think these values doesn't make any sense at all. Because we can't have any triangle with an angle of $400^{\circ}$ and we also can't have any sort of triangle with one angle being $0^{\circ}$ (or can we?). I am very confused with it. How can we find value of any trigonometric ratio for angles not in between $0^{\circ}$ & $90^{\circ}$ , if we can't make any triangle with these angles even in fantasy. So what are these $``0.64, -0.71, 1"?$

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We can think about trigonometric functions as those whose inputs are angles and outputs are scalars. For instance $sin(\theta)=x$. $\theta$ is the angle between your point of interest and 0 on the unit circle.

For the sine function the output scalar is the vertical component for the point on the unit circle which is an angle $\theta$ from 0. Alternatively cosine is the horizontal component.

We can talk about values with degree larger than what would make a triangle because we can keep looping around the unit circle. Thus values 0, 360, 720, etc obtain equal values when evaluated in trigonometric functions.

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