# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - **Maximum and Minimum Values of Trigonometric Functions**: Understanding how the amplitude and vertical shift affect the range of sine and cosine functions. - For $y = a + b \sin(\theta)$ or $y = a + b \cos(\theta)$, the maximum value is $a + |b|$ and the minimum value is $a - |b|$. - **Maximum and Minimum Values of Reciprocal Trigonometric Functions**: How the range of rational trigonometric functions depends on the range of their denominators. - If $y = \frac{1}{f(x)}$, then the maximum of $y$ is $1 / (\text{min of } f(x))$ and the minimum of $y$ is $1 / (\text{max of } f(x))$. - **Domain and Range of Trigonometric Functions**: Defining the permissible input values (domain) and the set of all possible output values (range) for different trigonometric functions (sine, cosine, tangent, cotangent). - **Even and Odd Functions**: Applying the definitions $f(-x) = f(x)$ for even functions and $f(-x) = -f(x)$ for odd functions to trigonometric expressions. - **Graphing Trigonometric Functions**: Understanding how amplitude, period, and phase shift transform the basic graphs of sine, cosine, and tangent. - **Amplitude**: Vertical stretch/compression. - **Period**: Horizontal stretch/compression ($T = \frac{2\pi}{|k|}$ for sine/cosine, $T = \frac{\pi}{|k|}$ for tangent/cotangent). - **Periods of Trigonometric Functions**: Calculating the period for individual functions and for sums/differences of functions using the Least Common Multiple (LCM) of individual periods. - **Graphical Solutions of Equations**: Using the intersection points of two graphs to find the solutions to an equation. - **Modeling with Sine Functions**: Applying trigonometric functions to real-world phenomena like alternating current (AC) to determine period, angular frequency, and amplitude. Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: 1. **Maximum and Minimum of Basic Trigonometric Functions** For a function of the form $y = a + b \sin(\theta)$ or $y = a + b \cos(\theta)$: $$ \text{Maximum: } M = a + |b| $$ $$ \text{Minimum: } m = a - |b| $$ 2. **Maximum and Minimum of Reciprocal Trigonometric Functions** For a function of the form $y = \frac{1}{f(x)}$: $$ \text{Maximum of } y: M_y = \frac{1}{\text{Minimum of } f(x)} $$ $$ \text{Minimum of } y: m_y = \frac{1}{\text{Maximum of } f(x)} $$ 3. **Period of Sine, Cosine, Secant, and Cosecant Functions** For functions $y = \sin(kx)$, $y = \cos(kx)$, $y = \sec(kx)$, or $y = \csc(kx)$: $$ \text{Period: } T = \frac{2\pi}{|k|} $$ 4. **Period of Tangent and Cotangent Functions** For functions $y = \tan(kx)$ or $y = \ cot(kx)$: $$ \text{Period: } T = \frac{\pi}{|k|} $$ 5. **Period of Combined Trigonometric Functions** For a sum or difference of trigonometric functions, the overall period is the Least Common Multiple (LCM) of their individual periods. 6. **Definition of an Even Function** A function $f(x)$ is even if: $$ f(-x) = f(x) $$ 7. **Definition of an Odd Function** A function $f(x)$ is odd if: $$ f(-x) = -f(x) $$ 8. **General Model for AC Voltage** The voltage can be modeled by the function: $$ V(t) = a \sin[k(t - d)] + c $$ where $a$ is the amplitude, $k$ is the angular frequency, $d$ is the phase shift, and $c$ is the vertical shift. 9. **Relationship Between Period and Frequency** If $f$ is the frequency: $$ \text{Period: } T = \frac{1}{f} $$ 10. **Relationship Between Angular Frequency and Frequency** If $f$ is the frequency: $$ \text{Angular Frequency: } k = 2\pi f $$ --- # **Summary** This exercise provided a comprehensive review of fundamental trigonometric function properties. Key topics covered include determining maximum and minimum values for various function types, identifying domains and ranges, classifying functions as even or odd, and interpreting graphical transformations related to amplitude and period. We also explored calculating periods for combined functions and applying trigonometric models to real-world phenomena like AC voltage, reinforcing practical problem-solving skills in trigonometry. --- # **Reference Videos** Below are recommended video explanations for this exercise: ---