Trigonometric functions are essential tools in calculus, representing ratios in right triangles and periodic behaviors. They're used to model everything from sound waves to planetary orbits. Understanding their properties and derivatives is crucial for solving real-world problems.

Knowing how to find the slopes and equations of tangent lines for trig functions opens up a world of applications. These skills help in optimization problems, like maximizing the area of shapes or analyzing oscillating systems in physics and engineering.

Trigonometric Functions and Their Derivatives

Basic trigonometric functions and properties

• Sine function $\sin(\theta)$ represents the ratio of the opposite side to the hypotenuse in a right triangle (e.g., $\sin(30°) = 0.5$)
  • Periodic function repeats every $2\pi$ radians (e.g., $\sin(0) = \sin(2\pi) = 0$)
  • Range limited to values between -1 and 1 inclusive (e.g., $\sin(\theta) \in [-1, 1]$)
• Cosine function $\cos(\theta)$ represents the ratio of the adjacent side to the hypotenuse in a right triangle (e.g., $\cos(60°) = 0.5$)
  • Periodic function repeats every $2\pi$ radians (e.g., $\cos(0) = \cos(2\pi) = 1$)
  • Range limited to values between -1 and 1 inclusive (e.g., $\cos(\theta) \in [-1, 1]$)
• Tangent function $\tan(\theta)$ represents the ratio of the opposite side to the adjacent side in a right triangle (e.g., $\tan(45°) = 1$)
  • Periodic function repeats every $\pi$ radians (e.g., $\tan(0) = \tan(\pi) = 0$)
  • Range includes all real numbers (e.g., $\tan(\theta) \in (-\infty, \infty)$)
• Reciprocal functions $\csc(\theta)$, $\sec(\theta)$, and $\cot(\theta)$ are the multiplicative inverses of sine, cosine, and tangent respectively
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$ (e.g., $\csc(90°) = 1$)
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$ (e.g., $\sec(0°) = 1$)
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$ (e.g., $\cot(45°) = 1$)

Derivative formulas for sine, cosine, and tangent

• Derivative of sine $\frac{d}{dx} \sin(x) = \cos(x)$ can be proven using the definition of the derivative and trigonometric identities (e.g., $\frac{d}{dx} \sin(\pi/6) = \cos(\pi/6) = \sqrt{3}/2$)
• Derivative of cosine $\frac{d}{dx} \cos(x) = -\sin(x)$ can be proven using the definition of the derivative and trigonometric identities (e.g., $\frac{d}{dx} \cos(\pi/4) = -\sin(\pi/4) = -\sqrt{2}/2$)
• Derivative of tangent $\frac{d}{dx} \tan(x) = \sec^2(x)$ is derived using the quotient rule $\frac{d}{dx} \tan(x) = \frac{d}{dx} \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \sec^2(x)$ (e.g., $\frac{d}{dx} \tan(\pi/3) = \sec^2(\pi/3) = 4$)

Applications of trigonometric derivatives

• Find the slope of the tangent line to a trigonometric function at a given point using the derivative formula (e.g., slope of $y=\sin(x)$ at $x=\pi/6$ is $\cos(\pi/6)=\sqrt{3}/2$)
• Determine the equation of the tangent line to a trigonometric function at a given point using the point-slope form $y - y_1 = m(x - x_1)$, where $m$ is the slope found using the derivative (e.g., tangent line to $y=\cos(x)$ at $x=\pi/4$ is $y - \frac{\sqrt{2}}{2} = \frac{-\sqrt{2}}{2}(x - \frac{\pi}{4})$)
• Solve optimization problems involving trigonometric functions:
  1. Set up the objective function and constraints using trigonometric functions
  2. Find the critical points by setting the derivative of the objective function equal to zero
  3. Evaluate the objective function at the critical points and endpoints to determine the optimal solution (e.g., maximize the area of a triangle inscribed in a unit circle)

Derivatives of trigonometric functions vs reciprocals

• Derivatives of reciprocal trigonometric functions involve the original function and its reciprocal:
  • $\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$ (e.g., $\frac{d}{dx} \csc(\pi/6) = -2\sqrt{3}$)
  • $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$ (e.g., $\frac{d}{dx} \sec(\pi/4) = \sqrt{2}$)
  • $\frac{d}{dx} \cot(x) = -\csc^2(x)$ (e.g., $\frac{d}{dx} \cot(\pi/3) = -4$)
• Compare derivatives of trigonometric functions and their reciprocals:
  • Derivatives of cosine and cotangent have negative signs (e.g., $\frac{d}{dx} \cos(x) = -\sin(x)$ and $\frac{d}{dx} \cot(x) = -\csc^2(x)$)
  • Derivatives of reciprocal functions involve both the original function and its reciprocal (e.g., $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$)