Pre-Calculus STEM 11 Q2 PDF

PRE-CALCULUS Grade 11 | 1st Semester | Quarter 2  Where i is the beginning, n is the last, and 𝑎𝑖 is the expression. Topic Outline:...

PRE-CALCULUS Grade 11 | 1st Semester | Quarter 2  Where i is the beginning, n is the last, and 𝑎𝑖 is the expression. Topic Outline:  Sequence & Series BINOMIAL EXPANSION o Sigma Notation Binomial Expansion – describes the algebraic expansion of powers o Arithmetic Sequence of a binomial (two-termed polynomial) o Geometric Sequence Blaise Pascal – a very influential French mathematician &  Binomial Expansion philosopher who contributed in many areas of mathematics o Pascal’s Triangle  Born: June 19, 1623 (Auvergne, France o Binomial Theorem Clermont/Clermont-Ferrand) – August 19, 1662 (Paris,  Angles & The Unit Circle France)  Unit Circle & The 6 Trigonometric Functions  His work: Pascaline, Pascal’s Mystic Hexagram, & Pascal Triangle SEQUENCE & SERIES PASCAL TRIANGLE Sequence – a set of related events, movements, or things that follow each other in a particular order & separated by commas (…, - A triangular arrangement of numbers that gives the coefficient I the infinite) expansion of any binomial expression  ARITHMETIC FORMULA: 𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅 o 𝑛 is the number of terms o 𝑑 is the common difference o 𝑎1 is the first term o 𝑎𝑛 is the last term  GEOMETRIC FORMULA: 𝒂𝒏 = 𝒂𝟏 𝒓𝒏−𝟏 o 𝑛 is the number of terms o 𝑟 is the common ratio (𝑟 ≠ 1) o 𝑎1 is the first term o 𝑎𝑛 is the last term (𝑥 + 𝑦)4 = 𝑥 4 + 4𝑥 3 𝑦 + 6𝑥 2 𝑦 2 + 4𝑥𝑦 3 + 𝑦 4 (2𝑥 − 1)3 = 8𝑥 3 − 12𝑥 2 + 6𝑥 − 1 Series – operation of adding infinitely many quantities; separated by 1 5 10 5 1 (𝑥 + ) = 𝑥 5 + 5𝑥 3 + 10𝑥 + + 3+ 5 plus or minus 𝑥 𝑥 𝑥 𝑥 𝒏 Note: If the middle sign is positive all terms are positive. If negative,  ARITHMETIC FORMULA: 𝑺𝒏 = 𝟐 (𝒂𝟏 + 𝒂𝒏 ) the sign of the terms are in an alternate scheme (+ & -) o 𝑛 is the number of terms o 𝑑 is the common difference 1. Expand (𝑥 + 𝑦)5 o 𝑎1 is the first term (𝒙 + 𝒚)𝟓 = 𝟏𝒙𝟓 𝒚𝟎 + 𝟓𝒙𝟒 𝒚𝟏 + 𝟏𝟎𝒙𝟑 𝒚𝟐 + 𝟏𝟎𝒙𝟐 𝒚𝟑 + 𝟓𝒙𝟏 𝒚𝟒 o 𝑎𝑛 is the last term 𝒂𝟏 (𝟏−𝒓𝒏 ) + 𝟏𝒙𝟎 𝒚𝟓  GEOMETRIC FORMULA: 𝑺𝒏 = 𝟏−𝒓 (𝒙 + 𝒚)𝟓 = 𝒙𝟓 + 𝟓𝒙𝟒 𝒚 + 𝟏𝟎𝒙𝟑 𝒚𝟐 + 𝟏𝟎𝒙𝟐 𝒚𝟑 + 𝟓𝒙𝒚𝟒 + 𝒚𝟓 o 𝑛 is the number of terms o 𝑟 is the common ratio (𝑟 ≠ 1) 2. Expand (𝟑𝒙 − 𝟐𝒚)𝟒 o 𝑎1 is the first term o 𝑎𝑛 is the last term (𝟑𝒙 − 𝟐𝒚)𝟒 = 𝟏(𝟑𝒙)𝟒 (𝟐𝒚)𝟎 − 𝟒(𝟑𝒙)𝟑 (𝟐𝒚)𝟏 + 𝟔(𝟑𝒙)𝟐 (𝟐𝒚)𝟐 SIGMA NOTATION −𝟒(𝟑𝒙)𝟏 (𝟐𝒚)𝟑 + 𝟏(𝟑𝒙)𝟎 (𝟐𝒚)𝟒 𝒏 (𝟑𝒙 − 𝟐𝒚)𝟒 = 𝟖𝟏𝒙𝟒 − 𝟐𝟏𝟔𝒙𝟑 𝒚 + 𝟐𝟏𝟔𝒙𝟐 𝒚𝟐 − 𝟗𝟔𝒙𝒚𝟑 + 𝟏𝟔𝒚𝟒 ∑ 𝒂𝒊 = 𝒂𝟏 + 𝒂𝟐 + 𝒂𝟑 + ⋯ 𝒂𝒏 𝒊=𝟏 1 PRE-CALCULUS Grade 11 | 1st Semester | Quarter 2 1 3 3. Expand (𝑥 − ) 𝑦 𝟏 𝟑 𝟏 𝟎 𝟏 𝟏 𝟏 𝟐 𝟏 𝟑 (𝒙 − ) = 𝟏(𝒙)𝟑 ( ) − 𝟑(𝒙)𝟐 ( ) + 𝟑(𝒙)𝟏 ( ) − 𝟏(𝒙)𝟎 ( ) 𝒚 𝒚 𝒚 𝒚 𝒚 𝟏 𝟒 𝟑𝒙𝟐 𝟑𝒙 𝟏 (𝒙 − ) = 𝒙𝟑 − + 𝟐− 𝟑 𝒚 𝒚 𝒚 𝒚 BINOMIAL THEOREM Clockwise direction – negative angle Standard Position – the vertex is at the origin & the initial side coincides with the positive x-axis o 𝒓 is the term o 𝒏 is the exponent Positive Side Negative Side o 𝒂 is the first term (binomial) Degree Measure – most commonly used unit of measure of an o 𝒃 is the second term (binomial) angle; 1o = 1 part of the 360 parts of the circle (To rad = divide it to 𝜋 180) 𝐷𝑒𝑔. 𝑡𝑜 𝑅𝑎𝑑. → 𝜃 × 180 1. Find the 5th term of (2𝑥 − 𝑦)8 Radian Measure – represent the amount of rotation of an angle; 𝒓𝒕𝒉 = 𝒏𝑪𝒓−𝟏 (𝒂)𝒏−𝒓+𝟏 (𝒃)𝒓−𝟏 one radian is the measure of a central angle of a circle (To deg. = 180 multiply it to 180) 𝑅𝑎𝑑. 𝑡𝑜 𝐷𝑒𝑔. → 𝜃 × 𝜋 5𝑡ℎ = 8𝐶5−1 (2𝑥)8−5+1 (−𝑦)5−1 Coterminal Angles – are angles having the same initial & terminal 5𝑡ℎ = 8𝐶4 (2𝑥)4 (−𝑦)4 sides; add & subtract 360 5𝑡ℎ = 70(16𝑥 4 )(𝑦 4 ) Reference Angle – defined as the acute angle & it is always positive measuring less than 90 degrees 5𝑡ℎ = 𝟏𝟏𝟐𝟎𝒙𝟒 𝒚𝟒 ANGLES & THE UNIT CIRCLE Angle – formed when two straight lines or rays meet at a common endpoint (vertex) UNIT CIRCLE & THE 6 TRIGONOMETRIC Terminal side – ray that is rotated FUNCTIONS Initial side – the position Unit Circle – a circle with radius equals to 1 unit & is centered at the to which the rotation origin (0,0); For each coordinates: (𝒙, 𝒚) = (𝐜𝐨𝐬 𝜽 , 𝐬𝐢𝐧 𝜽) ends Counterclockwise direction – positive angle 2 PRE-CALCULUS Grade 11 | 1st Semester | Quarter 2 CIRCULAR FUNCTIONS Circular Function – referred to as the functions determined by the length of the arcs on a circle TERMS: 𝜃 → angle in degrees; 𝑠 → angle in radians COORDINATES: (𝑥, 𝑦) → (cos 𝜃 , sin 𝜃)or (cos 𝑠 , sin 𝑠) 𝜋 MULTIPLIERS: 𝐷𝑒𝑔. 𝑡𝑜 𝑅𝑎𝑑. → 𝜃 × 180 180 𝑅𝑎𝑑. 𝑡𝑜 𝐷𝑒𝑔. → 𝜃 × 𝜋 CIRCULAR FUNCTION: 6 TRIGONOMETRIC 𝒚 𝐬𝐢𝐧 𝜽 = 𝐬𝐢𝐧 𝒔 = (sine) 𝒓 𝒙 𝐜𝐨𝐬 𝜽 = 𝐜𝐨𝐬 𝒔 = (cosine) 𝒓 𝒚 𝐭𝐚𝐧 𝜽 = 𝐭𝐚𝐧 𝒔 = (tangent) 𝒙 𝒓 𝐜𝐬𝐜 𝜽 = 𝐜𝐬𝐜 𝒔 = (cosecant) 𝒚 𝒓 𝐬𝐞𝐜 𝜽 = 𝐬𝐞𝐜 𝒔 = (secant) 𝒙 𝒙 𝐜𝐨𝐭 𝜽 = 𝐜𝐨𝐭 𝒔 = (cotangent) 𝒚 𝒓 = √𝒙𝟐 + 𝒚𝟐 𝒙 = √𝒓𝟐 − 𝒚𝟐 𝒚 = √𝒓𝟐 − 𝒙𝟐 3