Where is cotangent not defined

But sine itself, would not be invertible because it's not injective, so it's not bijective invertible. What is unit circle in math? In mathematics, a unit circle is a circle with unit radius. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin 0, 0 in the Cartesian coordinate system in the Euclidean plane.

What is a reference angle? The reference angle is the positive acute angle that can represent an angle of any measure. The reference angle is always the smallest angle that you can make from the terminal side of an angle ie where the angle ends with the x-axis.

How many radians are in a circle? Is CSC the inverse of sin? The cosecant is the reciprocal of the sine; the arcsin of x is the angle whose sine is x. What is the ASTC rule? The abbreviation for 'all sin cos tan' rule in trigonometry is ASTC.

The first letter of the third word T indicates that tangent and its reciprocal is positive in the third quadrant. The first letter of the last word C indicates that cosine and its reciprocal are positive in the fourth quadrant. Remember that the tangent function can be defined either as the quotient of the opposite and adjacent, or as the quotient of the sine and cosine functions. As mentioned, the tangent function is the quotient of the sine and cosine functions.

Since the cotangent function is the reciprocal of the tangent function, we can also express the cotangent function in terms of the sine and cosine function:. Obviously, since the cotangent function is the reciprocal of the tangent function, it can be expressed in terms of the tangent function as:.

And finally, since the cosine and sine functions are represented by the x and y coordinates respectively of point P i. We can also define the cotangent function in terms of a right-angled triangle.

If you have read the pages in this section covering the three main trigonometric functions sine , cosine and tangent you will by now be familiar with the diagram below.

This gives us:. To see the table, click here. In fact, it will apply to any angle for which the tangent value is zero. Like the tangent function, the values of the cotangent function will vary between zero and infinity or more correctly, between zero and a very large undefined value.

Because the cotangent function is the reciprocal of the tangent function, the cotangent value will be undefined when the tangent value is zero, and zero when the tangent value is undefined. As you can see by studying the table of cotangent values and the graph of the cotangent function, the cotangent value is negative for all angles for which the sine and cosine have a different sign i. If you have read the pages on tangents, cosecants and secants or possibly even if you haven't you will know that the vertical red lines on the graph are asymptotes.

The asymptote of a curve is a line such that as the distance between the curve and the line approaches zero as the value of the curve approaches infinity.

You can also see how the values differ on either side of the asymptote by studying the table of cotangent values. You might recall that the tangent function returns a value of zero at zero degrees, and rises to its positive peak as it approaches ninety degrees.

It then transitions to its negative peak, returning to zero at one hundred and eighty degrees before repeating the cycle again. The cotangent function is like a mirror image of the tangent function, phase shifted ninety degrees left or right. It falls from its positive peak as it moves away from zero degrees to a value of zero at ninety degrees , and then falls again to its negative peak as it approaches one hundred and eighty degrees. Since the ratio of any two real numbers is undefined when the denominator is equal to , must be undefined for those values of where.

Restricting our attention to those values of between and , when or. Hence, is undefined when or. All negative numbers and 0. All positive numbers and 0. All real numbers except 0.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f x. Trigonometric functions are undefined when they represent fractions with denominators equal to zero. In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. Which of the following trigonometric functions is undefined? The value of tan pi is 0, so the cotangent of pi must be undefined.

The value of tan 0 is 0, so the cotangent of 0 must be undefined. The value of sin 0 is 0, so the cosecant of 0 must be undefined. In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle.

The value of sin pi is 0, so the cosecant of pi must be undefined. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

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