The Derivative of a Sine Trigonometric Function
In order to easily compute the derivative of Sin 2X one must understand how the trigonometric impressions of the function Sine. This serves as the first important approach that helps one discretely complement Sine with its ‘sister’ Cosine. Such impressions can easily improve one’s comprehension of the changes in Sine as the angle ‘X’ changes from 0o to 3600. It is from this understanding that the intricate changes between this trigonometric impressions that one is able to effectively understand how the derivative of Sin 2X is obtained. This includes how such derivatives change with the change in X.
First things First: Understanding Sine Functions as Featured on the Cartesian plane
Let us begin by first understanding how the changes in various constants can affect the orientation of a Sine curve over the universal Cartesian plane. The following figure depicts how changes in magnitude on constant a affects the changes on the Sine function depicted on the graph. However, when it comes to computing the derivative of a Sin curve, despite the changes in magnitude as a result of changes in a, these do not affect the value of the resultant asymptote as depicted on both graphs below. In fact, it is for this same reason that the value for Sin 2X must indeed have a different asymptote as opposed to Sin X (that is, the latter is displaced vertically when compared to the former). However, this was not the reason of employing these graphs below but only to serve as a way of proving how the Sine Functions change to Tan Functions whenever one transforms the former into the latter.
It is a general mathematical law that the rate of change of a variable in relation to another can be computed, albeit graphically, by the use of tangents. Tangents are more accurately termed as asymptotes especially when trying to determine the most infinitesimal changes of one variable to another. It is at this approach to asymptotes that one commonly hears the phrases “as the limit tends to” or “as the limit tends towards zero.” In this case, the case if always “as the limit tends towards zero” since we are trying to determine the exact location of x (as depicted on the graph above) as this variable is subjected to various other potentially transformative variables such as a, b, h and k (also depicted above).
The Actual Computation of the Derivative of Sin 2X
STEP 1: Understanding the Algebra behind Trigonometric Ratios
Select two arbitrary variables; x and y. Determine the first quotient of the difference between these two variables leads us to the realization of Sin (x+h). However, you must note that the figure above uses the variable h in place of y for this first step. Using the angle addition formula, we find that Sin (x +y) can be expressed as:
Upon further simplification, the above equation reduces to:
Obtaining the limits of this equation as h tends to zero is the actual computation that leads to the realization of trigonometric ratios capable of facilitating the solution of the derivative of Sin2x. This is done for both 1 and 0 respectively.
The answer to this limit functions forms the vital trigonometric ratios and functions whose mastery is critical to understanding how derivatives for trigonometric angles are obtained.
Step 2: An Example Depiction of Definitive Derivatives of Discrete Trigonometric Functions
STEP 3: Performing the Actual Transformation of Sin2x into its Derivative
In this final step, it is important to learn that computational ‘differentiation’ on any particular trigonometric function starkly differs for circumstances where computational ‘derivation’ is performed. Whereas the former has assigned a predetermined arbitrary variable, say y whose change is evaluated (as is the case with dy/dx), the latter only employs changes in relation to angles (that is d/dƟ).
Using the chain rule we have (just as a prove that Steps 1 and 2 as outlined above are critical to this derivation task):
This rule relies on the dot cross between cosine and sine functions as follows:
- Assign a function to sin2x , say f(x) whilst paying attention to their domain
Then the computation becomes;
Thus the derivative of Sin2 x is Cos2 x obtained by using the distribution law to obtain the dot cross multiplication answer above.
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