Derivative of Sine, Cosine, and Tangent: Differentiating Trigonometric Functions

Example 1:

A curve has equation \( y = 3 \cos 2x + 4 \sin 2x + 1 \) for \( 0 \leq x \leq \pi \). Find the coordinates of the stationary points of the curve, correct to 3 significant figures.

Solution:

To find the stationary points, we need to differentiate \( y \) with respect to \( x \) and set \( \frac{dy}{dx} = 0 \).

We have:

\[ y = 3 \cos 2x + 4 \sin 2x + 1 \]

Differentiate \( y \):

\[ \frac{dy}{dx} = -6 \sin 2x + 8 \cos 2x \]

Set \( \frac{dy}{dx} = 0 \):

\[ -6 \sin 2x + 8 \cos 2x = 0 \] \[ 8 \cos 2x = 6 \sin 2x \] \[ \tan 2x = \frac{4}{3} \]

Solving for \( x \) in the interval \( 0 \leq x \leq \pi \):

\[ 2x = \tan^{-1} \frac{4}{3} \] \[ x \approx 0.464 \quad \text{and} \quad x \approx 2.03 \]

Substitute these \( x \)-values back into the original equation to find the corresponding \( y \)-coordinates:

\[ y(0.464) \approx 3 \cos(2 \cdot 0.464) + 4 \sin(2 \cdot 0.464) + 1 \] \[ = 3 \cos(0.928) + 4 \sin(0.928) + 1 \] \[ \approx 3(0.599) + 4(0.801) + 1 = 0.464 \]