Show that (d)/(d x) cos(x^∘) is -(π)/(180) sin(x^∘). |...

Name: Problem 57, Chapter 4: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2021-09-01T14:34:29-08:00
Duration: 4 min 28 s
Channel: Andrija Isakov
Description: Show that (d)/(d x) cos(x^∘) is -(π)/(180) sin(x^∘).

Key Concepts

Differentiation of Trigonometric Functions

Differentiating trigonometric functions is a core topic in calculus. The derivative of cos(x) with respect to x is -sin(x). When the input is modified, as in cos(x°) with the degree to radian conversion, the standard derivative formula is adjusted by the derivative of the conversion function. This leads directly to the derivative being multiplied by the extra factor, resulting in the derivative - (?/180)·sin(x°).

Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. It states that if you have a function of the form f(g(x)), then the derivative is the product of the derivative of f evaluated at g(x) and the derivative of g. In this problem, the chain rule is applied because the cosine function is composed with a conversion function that changes the input from degrees to radians.

Conversion Between Degrees and Radians

In calculus, trigonometric functions assume that the angle is measured in radians. However, in many practical problems, angles might be given in degrees. To correctly differentiate these functions, it is essential to convert degrees to radians using the relationship 1° = ?/180 radians. This conversion introduces the factor of ?/180 in the derivative when using the chain rule.

Transcript

00:01 In this question, we want to show that the derivative of cosine of x degrees is equal to minus pi over 180 sine of x degrees.

00:13 So, first of all, how do we take derivatives of cosine functions? well, we know cosine of just the variable x, which means x measured in radiance.

00:28 So, cosine of the raw number x.

00:34 We know the derivative of that is minus sine of x.

00:39 And also if we had cosine of a function of x, then the derivative of that, by the chain rule, is equal to the derivative of the cosine part with the exact same argument, times the derivative of the argument with respect to x.

01:08 In a sense, the derivative of the outside times the derivative of the inside.

01:16 So how do we use that to differentiate cosine of x degrees? degrees, sorry.

01:24 Well, we can express x degrees as a raw number, like we saw with cosine of x.

01:33 Now, cosine of x degrees is not exactly cosine of x, because cosine of x is, x means x radians, and a radian is about this much of a circle, whereas a degree is much smaller.

01:51 And so x degrees is a lot different than x radiance.

01:57 They do, however, have a direct relationship.

02:01 And that's most easily seen using a half circle.

02:09 So if we have, if you imagine the usual unit circle diagram on the x, y plane, and if we go half a circle, that's going to give us 180 degrees...

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