2.3: Derivatives of Trigonometric Functions (Lecture Notes)
- Page ID
- 122398
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Theorems
The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.\[\dfrac{d}{dx}(\sin{(x)})=\cos{( x)} \nonumber \]\[\dfrac{d}{dx}(\cos{(x)})=−\sin{( x)} \nonumber \]
- Proof
-
\[\begin{array}{rclcl}
\dfrac{d}{dx}(\sin x) & = & \displaystyle \lim_{h \to 0}\dfrac{\sin(x+h)−\sin x}{h} & \quad & \left( \text{definition of the derivative} \right) \\[16pt]
& = & \displaystyle \lim_{h \to 0}\dfrac{\sin x\cos h+\cos x\sin h−\sin x}{h} & \quad & \left( \text{Sum of Angles Identity} \right) \\[16pt]
& = & \displaystyle \lim_{h \to 0}\left(\dfrac{\sin x\cos h−\sin x}{h}+\dfrac{\cos x\sin h}{h}\right) & \quad & \left( \text{regrouping} \right) \\[16pt]
& = & \displaystyle \lim_{h \to 0}\left(\sin x\left(\dfrac{\cos h−1}{h}\right)+(\cos x)\left(\dfrac{\sin h}{h}\right)\right) & \quad & \left( \text{factoring} \right) \\[16pt]
& = & (\sin x)\lim_{h \to 0}\left(\dfrac{\cos h−1}{h}\right)+(\cos x)\lim_{h \to 0}\left(\dfrac{\sin h}{h}\right) & \quad & \left( \text{factoring} \right) \\[16pt]
& = & (\sin x)(0)+(\cos x)(1) & & \\[16pt]
& = & \cos x & \\[16pt]
\end{array} \nonumber \]
The derivatives of the remaining trigonometric functions (along with the sine and cosine) are as follows:\[ \begin{array}{rclcrcl}
\dfrac{d}{dx}(\sin{(x)}) & = & \cos{(x)} & \quad & \dfrac{d}{dx}(\csc{(x)}) & = & −\csc{(x)} \cot{(x)} \\
\dfrac{d}{dx}(\cos{(x)}) & = & -\sin{(x)} & \quad & \dfrac{d}{dx}(\sec{(x)}) & = & \sec{(x)} \tan{(x)} \\
\dfrac{d}{dx}(\tan{(x)}) & = & \sec^2{(x)} & \quad & \dfrac{d}{dx}(\cot{(x)}) & = & −\csc^2{(x)} \\
\end{array} \nonumber \]
Examples
Differentiate.
- \( f(x) = \frac{1}{\sin(x) + \tan(x)} \)
- \( g(\theta) = \frac{\cot(\theta)}{\cos(\theta)} + \sqrt{\theta} \sec(\theta) \)
Let \( h(x) = \tan(x) - 2 \sin(x) \). Find the equation of the tangent line to this curve at \( x = \pi \)
Find the \(310^{\text{th}}\) derivative of \(f(x)=x \cos(x)\).
Find constants \(A\) and \(B\) such that the function \(y = A \sin(x) + B \cos(x)\) satisfies the differential equation \(y^{\prime\prime} + y^{\prime} - 2 y = \sin(x)\).
Compute the following.\[ \dfrac{d^2}{dx^2} \left[ \sec^2(x) \right] \nonumber \]