Trigonometric Functions and Their Properties LEC3✅

  • Trigonometric Functions Introduction

    • Focus on sine (sin) and cosine (cos).
    • Use of the unit circle for definitions.

  • Unit Circle

    • Radius = 1; defines x and y coordinates.
    • Cosine: x-coordinate = adjacent side; $\cos(x) = \frac{adjacent}{hypotenuse}$ (hypotenuse = 1).
    • Sine: y-coordinate = opposite side; $\sin(x) = \frac{opposite}{hypotenuse}$ (hypotenuse = 1).
    • Identity: $\cos^2(x) + \sin^2(x) = 1$.

  • Angle Measurement

    • Angles measured in radians (easier for calculus).
    • $2\pi$ radians = 360 degrees.
    • Principal angles between $-\pi$ and $\pi$.

  • Periodic Functions

    • Sine and cosine are periodic with a period of $2\pi$.
    • $\sin(-x) = -\sin(x)$ (odd function); $\cos(-x) = \cos(x)$ (even function).

  • Amplitude and Phase Shift

    • General form: $y = a \sin(kx + \alpha)$ or $y = a \cos(kx + \alpha)$.
    • Amplitude (a) is always positive.
    • Phase shift (α) determines the horizontal shift of the function.

  • Tangent Function

    • Defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$.
    • Domain excludes values where $\cos(x) = 0$ (vertical asymptotes).
    • Odd function: $\tan(-x) = -\tan(x)$.

  • Inverse Trigonometric Functions

    • Restrict tan function to $(-\frac{\pi}{2}, \frac{\pi}{2})$ for the inverse to exist.
    • Domain of $\arctan(x)$ is all real numbers, range is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

  • Key Concepts to Remember

    • $\cos^2(x) + \sin^2(x) = 1$.
    • Understand the periodicity of sine and cosine functions.
    • Know the relationship between degrees and radians: $1 \text{ radian} = \frac{180}{\pi} \text{ degrees}$.