📂Functions

The Relationship between the Translation of Trigonometric Functions and Their Derivatives

Formulas

• [1] Sine: $$\sin{(\theta +\frac { n }{ 2 }\pi )}={ \sin }^{ (n) }\theta$$
• [2] Cosine: $$\cos{(\theta +\frac { n }{ 2 }\pi )}={ \cos }^{ (n) }\theta$$

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• $(n)$ means it is differentiated $n$ times.

Explanation

Simply put, differentiate once every time you move 90˚. Let’s actually calculate for $n=3$.

Method using the Addition Theorem

\begin{align*} \cos(\theta +{3 \over 2}\pi ) =& \cos\theta \cos\frac { 3 }{ 2 }\pi -\sin\theta \sin\frac { 3 }{ 2 }\pi \\ =& \cos\theta \cdot 0-\sin\theta \cdot (-1) \\ =& \sin\theta $ \end{align*}

공식을 사용한 방법

$$ \begin{align*} { \cos }^{ (3) }\theta =& (\cos\theta )’’’ \\ =& (-\sin\theta )’’ \\ =& (-\cos\theta )’ \\ =& \sin\theta \end{align*} $Though shifting by 90˚ cases aren’t given often, it’s still a very simple formula, so instead of memorizing it, having it in mind will be helpful.