Pre-Calculus: Sequences, Series, and Angles PDF

# Pre-Calculus: Sequences, Series, and Angles ## Sequences - A set of numbers with a pattern. - Each number is called a "term". - Must be in orderly fashion. ### Types of Sequences - **Fibonacci:** next term is found by adding the two previous terms. - Example: 0, 1, 1, 2, 3, 5... - For...

# Pre-Calculus: Sequences, Series, and Angles ## Sequences - A set of numbers with a pattern. - Each number is called a "term". - Must be in orderly fashion. ### Types of Sequences - **Fibonacci:** next term is found by adding the two previous terms. - Example: 0, 1, 1, 2, 3, 5... - Formula: $A_n = A_{n-1} + A_{n-2}$ - **Arithmetic:** common difference. - Example: 2, 5, 8, 11, 14... (d = 3) - Formula: $A_n = A_1 + (n - 1)d$ - **Geometric:** common ratio. - Example: 3, 6, 12,24, 48... (r = 2) - Formula: $A_n = A_1 (r^{n-1})$ ## Series - The sum of the terms of a sequence. - Example: 1, 3, 5, 7, 9, 11... - Series: 1 + 3 + 5 + 7 + 9 + 11 ## Sigma Notation - Simplest form of shorthand used to give a concise expression for a series. - Notation: * $\sum\limits_{k = 1}^n a_k = a_1 + a_2 +...+ a_{n-1} + a_n$ ## Properties of Sigma Notation 1. $\sum\limits_{k=1}^n c = nc$ (c is constant) 2. $\sum\limits_{k=1}^n c(a_k) = c (\sum\limits_{k=1}^n a_k)$ 3. $\sum\limits_{k=1}^n (a_k + b_k) = \sum\limits_{k=1}^n (a_k) + \sum\limits_{k=1}^n (b_k)$ ## Summation Formula 1. $\sum\limits_{k=1}^n k = \frac{n(n + 1)}{2}$ 2. $\sum\limits_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}$ 3. $\sum\limits_{k=1}^n k^3 = \frac{n^2(n + 1)^2}{4}$ ## Arithmetic and Geometric Sequences and Series ### Arithmetic Series - General formula: $a_n = \frac{n}{2}(a_1 + a_n)$ ### Geometric Series - General formula: $G_n = a_1 \frac{(1 - r^n)}{(1 - r)}$ - Infinite geometric series: $G_n = \frac{a_1}{1 - r}$ if $|r| < 1$ ## Angle and Angle Measure - **Trigonometry:** Mathematics of angles. From Greek, trigonon = three angles, metron = to measure. - **Hipparchus of Nicaea:** Father of trigonometry, first compiled trigonometric values. - **Angle:** Amount of rotation between two lines/rays/line segments having a common endpoint formed by two rays originating from the vertex. ### Classification of Angles: 1. **Acute angle:** Less than 90°. 2. **Right angle:** Exactly 90°. 3. **Obtuse angle:** More than 90° but less than 180°. 4. **Straight angle:** Exactly 180°. 5. **Reflex angle:** Greater than 180° but less than 360°. 6. **Full rotation:** Exactly 360°. ### Central Angle and Arc - **Central angle:** Angle inside the circle with its vertex at the center of the circle. - **Arc:** Portion of the circumference of the circle formed from the endpoints of the angle. ### Angle Measure - Radian measurement is based on the radius of the circle: - $1^\circ = (\frac{\pi}{180}) \text{ rad}$ - $\text{ rad} = (\frac{180}{\pi})^\circ$ - Arc Length: - $s = \theta r$ * $s$ = length of arc * $\theta$ = measure of the central angle in radians * $r$ = length of the radius ## Angle in Standard Position - An angle where one side is located on the x-axis, with the vertex at the origin. - **Coterminal Angles:** Angles in standard position having the same terminal sides. These can be found by adding or subtracting multiples of 360°. ## Reference Angles - The smallest positive angle between the terminal side of an angle in standard position and the x-axis. - **Finding a Reference Angle:** - **Quadrant I:** $r = $ given angle - **Quadrant II:** $r = 180^\circ - $ given angle - **Quadrant III:** $r = $ given angle $- 180^\circ$ - **Quadrant IV:** $r = 360^\circ - $ given angle - **Important Note:** If the given angle is negative, switch the sign of the reference angle. ## Converting Between Degrees and Radians - Degrees to radians: $(\text{degree}) * (\frac{\pi}{180})$ - Radians to degrees: $(\text{radian}) * (\frac{180}{\pi})$ ## Finding Coterminal Angles - **To find a positive coterminal angle:** Add 360° repeatedly to the given angle. - **To find a negative coterminal angle:** Subtract 360° repeatedly from the given angle.