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In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side of that angle to the hypotenuse.

Q: How is cosine defined?

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That is how the scalar product is defined. Also, the projection of one vector onto another at an angle to it is directly proportional to the cosine of that angle.

Tangent (theta) is defined as sine (theta) divided by cosine (theta). In a right triangle, it is also defined as opposite (Y) divided by adjacent (X).

Tan of pi/2 + k*pi radians, for integer k, is not defined since tan = sin/cos and the cosine of these angles is 0. Since divsiion by 0 is not defined, the tan ratio is not defined.

The range of cosine is [-1, 1] which is, therefore, the domain of cos-1. As a result, cos-1(2) is not defined.

what is the cosine of 3.14 ?

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The secant of an angle is the reciprocal of the cosine of the angle. So the secant is not defined whenever the cosine is zero That is, whenever the angle is a multiple of 180 degrees (or pi radians).

Yes, except at odd multiples of pi/2 radians, where the cosine is zero so that the division is not defined.

That is how the scalar product is defined. Also, the projection of one vector onto another at an angle to it is directly proportional to the cosine of that angle.

It is cosine*cosine*cosine.

Tangent (theta) is defined as sine (theta) divided by cosine (theta). In a right triangle, it is also defined as opposite (Y) divided by adjacent (X).

Tan of pi/2 + k*pi radians, for integer k, is not defined since tan = sin/cos and the cosine of these angles is 0. Since divsiion by 0 is not defined, the tan ratio is not defined.

The tangent of an angle theta is defined as sine(theta) divided by cosine(theta). Since the sine and cosine are Y and X on the unit circle, then tangent(theta) is Y divided by X. The tangent of a function at a point is the line going through that point which has slope equal to the first deriviative of the function at that point.

The cosine function, like all of the trigonometric functions, is periodic about the rotation around a circle. Since the cosine is defined as the adjacent/hypotenuse of a right triangle, you can clearly see that its value can never be greater than one or less than -1 since the hypotenuse is always longer than the adjacent side. It turns out that, indeed, the cosine's range is from -1 to 1, written [-1,1].

The range of cosine is [-1, 1] which is, therefore, the domain of cos-1. As a result, cos-1(2) is not defined.

Cosine 0 is 1

what is the cosine of 3.14 ?

Cosine of 1 degree is about 0.999848. Cosine of 1 radian is about 0.540302.