precalculus formulas cheat sheet, Cheat Sheet of Mathematics

Download precalculus formulas cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2018Fall  System of Equations and Matrices  3 Matrix Row Operations: • Switch any two rows. • Multiply any row by a nonzero constant. • Add any constant-multiple row to another  Even and Odd functions • Even function: 𝑓(−𝑥) = 𝑓(𝑥) Odd function: 𝑓(−𝑥) = −𝑓(𝑥)  Graph Symmetry • 𝑥-axis symmetry: if (𝑥, 𝑦) is on the graph, then (𝑥, −𝑦) is also on the graph • 𝑦-axis symmetry: if (𝑥, 𝑦) is on the graph, then (−𝑥, 𝑦) is also on the graph • origin symmetry: if (𝑥, 𝑦)f is on the graph, then (−𝑥, −𝑦) is also on the graph  Function Transformations  Stretch and Compress • 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1  Reflections • 𝑦 = −𝑓(𝑥) reflect 𝑓(𝑥) about 𝑥-axis • 𝑦 = 𝑓(−𝑥) reflect 𝑓(𝑥) about 𝑦-axis  Stretch and Compress • 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1 : compress 𝑓(𝑥) if 0 < 𝑎 < 1 • 𝑦 = 𝑓(𝑎𝑥), 𝑎 > 0 horizontal: stretch 𝑓(𝑥) if 0 < 𝑎 < 1 : compress 𝑓(𝑥) if 𝑎 > 1  Shifts • 𝑦 = 𝑓(𝑥) + 𝑘, 𝑘 > 0 vertical: shift 𝑓(𝑥) up 𝑦 = 𝑓(𝑥) − 𝑘, 𝑘 > 0 : shift 𝑓(𝑥) down • 𝑦 = 𝑓(𝑥 + ℎ) ℎ > 0 horizontal: shift 𝑓(𝑥) left 𝑦 = 𝑓(𝑥 − ℎ), ℎ > 0 : shift 𝑓(𝑥) right CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2018Fall  Formulas/Equations • Slope Intercept: 𝑦 = 𝑚𝑥 + 𝑏 Point-Slope: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) • Slope: 𝑚 = 𝑦2−𝑦1 𝑥2−𝑥1 ; 𝑥2 − 𝑥1 ≠ 0 • Average Rate of Change: Δ𝑦 Δ𝑥 = 𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎 , where 𝑎 ≠ 𝑏 • Circle: 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋𝑟 = 𝜋𝑑, 𝐴𝑟𝑒𝑎 = 𝜋𝑟2 • Triangle: 𝐴𝑟𝑒𝑎 = 1 2 𝑏ℎ • Rectangle: 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑙 + 2𝑤 , 𝐴𝑟𝑒𝑎 = 𝑙𝑤 • Rectangular Solid: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑤ℎ, 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ • Sphere: 𝑉𝑜𝑙𝑢𝑚𝑒 = 4 3 𝜋𝑟3 , 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 4𝜋𝑟2 • Right Circular Cylinder: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟2ℎ , 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝜋𝑟2 + 2𝜋𝑟ℎ  General Form of Quadratic Function: 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 , (𝑎 ≠ 0) • Quadratic Formula: 𝑥 = −𝑏±√𝑏2−4𝑎𝑐 2𝑎 • Vertex (ℎ, 𝑘): ℎ = − 𝑏 2𝑎 𝑘 = 𝑎(ℎ)2 + 𝑏(ℎ) + 𝑐, or (− 𝑏 2𝑎 , 𝑓 (− 𝑏 2𝑎 )), or (− 𝑏 2𝑎 , 4𝑎𝑐−𝑏2 4𝑎 ) • Axis of symmetry: 𝑥 = ℎ  Vertex Form of Quadratic Function: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 vertex (ℎ, 𝑘)  Polynomial function: 𝑓(𝑥) = 𝑎𝑛𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 + ⋯ + 𝑎1𝑥 1 + 𝑎0  Polynomial graph has at most 𝑛 − 1 turning points.  Remainder Theorem • If polynomial 𝑓(𝑥) ÷ (𝑥 − 𝑐), remainder is 𝑓(𝑐).  Factor Theorem • If 𝑓(𝑐) = 0, then 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥). • If 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥), then 𝑓(𝑐) = 0.  Rational Zeros Theorem • Possible rational zeros: ± 𝑝 𝑞 , where 𝑝 is a factor of 𝑎0 and 𝑞 is a factor of 𝑎𝑛. CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2018Fall  Trigonometry  Circular Measure and Motion Formulas • Arc Length 𝑠 = 𝑟𝜃 Area of Sector 𝐴 = 1 2 𝑟2𝜃 • Linear Speed 𝑣 = 𝑠 𝑡 , 𝑣 = 𝑟𝜔 Angular Speed 𝜔 = 𝜃 𝑡  Acute Angle • sin(𝜃) = 𝑏 𝑐 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos(𝜃) = 𝑎 𝑐 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan(𝜃) = 𝑏 𝑎 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 • csc(𝜃) = 𝑐 𝑏 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 sec(𝜃) = 𝑐 𝑎 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 cot(𝜃) = 𝑎 𝑏 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒  General Angle • sin(𝜃) = 𝑏 𝑟 cos(𝜃) = 𝑎 𝑟 tan(𝜃) = 𝑏 𝑎 • csc(𝜃) = 𝑟 𝑏 ,𝑏 ≠ 0 sec(𝜃) = 𝑟 𝑎 ,𝑎 ≠ 0 cot(𝜃) = 𝑎 𝑏 ,𝑏 ≠ 0  Cofunctions • sin(𝜃) = cos ( 𝜋 2 − 𝜃) , cos(𝜃) = sin ( 𝜋 2 − 𝜃) , tan(𝜃) = cot ( 𝜋 2 − 𝜃) • csc(𝜃) = sec ( 𝜋 2 − 𝜃) , sec(𝜃) = csc ( 𝜋 2 − 𝜃) , cot(𝜃) = tan ( 𝜋 2 − 𝜃)  Fundamental Identities • tan(𝜃) = sin(𝜃) cos(𝜃) cot(𝜃) = cos(𝜃) sin(𝜃) • csc(𝜃) = 1 sin(𝜃) sec(𝜃) = 1 cos(𝜃) cot(𝜃) = 1 tan(𝜃) • sin2(𝜃) + cos2(𝜃) = 1 tan2(𝜃) + 1 = sec2(𝜃) cot2(𝜃) + 1 = csc2(𝜃)  Even-Odd Identities • sin(−𝜃) = −sin(𝜃) cos(−𝜃) = cos(𝜃) tan(−𝜃) = − tan(𝜃) • csc(−𝜃) = −csc(𝜃) sec(−𝜃) = sec(𝜃) cot(−𝜃) = − cot(𝜃)  Inverse Functions • 𝑦 = sin −1(𝑥) means 𝑥 = sin (𝑦) where −1 ≤ 𝑥 ≤ 1 and − 𝜋 2 ≤ 𝑦 ≤ 𝜋 2 • 𝑦 = cos −1(𝑥) means 𝑥 = cos (𝑦) where −1 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑦 ≤ 𝜋 • 𝑦 = tan −1(𝑥) means 𝑥 = tan (𝑦) where −∞ ≤ 𝑥 ≤ ∞ and − 𝜋 2 < 𝑦 < 𝜋 2 • 𝑦 = csc −1(𝑥) means 𝑥 = csc (𝑦) where |𝑥| ≥ 1 and − 𝜋 2 ≤ 𝑦 ≤ 𝜋 2 , 𝑦 ≠ 0 • 𝑦 = sec −1(𝑥) means 𝑥 = sec (𝑦) where |𝑥| ≥ 1 and 0 ≤ 𝑦 ≤ 𝜋, 𝑦 ≠ 𝜋 2 • 𝑦 = cot −1(𝑥) means 𝑥 = cot (𝑦) where −∞ ≤ 𝑥 ≤ ∞ and 0 < 𝑦 < 𝜋 CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2018Fall  Sum and Difference Formulas • sin(𝛼 + 𝛽) = sin(𝛼) cos(𝛽) + cos(𝛼) sin(𝛽) • sin(𝛼 − 𝛽) = sin(𝛼) cos(𝛽) − cos(𝛼) sin(𝛽) • cos(𝛼 + 𝛽) = cos(𝛼) cos(𝛽) − sin(𝛼) sin(𝛽) • cos(𝛼 − 𝛽) = cos(𝛼) cos(𝛽) + sin(𝛼) sin(𝛽) • tan(𝛼 + 𝛽) = tan(𝛼)+tan(𝛽) 1−tan(𝛼) tan(𝛽) tan(𝛼 − 𝛽) = tan(𝛼)−tan(𝛽) 1+tan(𝛼) tan(𝛽)  Half-Angle Formulas • sin ( 𝛼 2 ) = ±√ 1−cos(𝛼) 2 • cos ( 𝛼 2 ) = ±√ 1+cos(𝛼) 2 • tan ( 𝛼 2 ) = ±√ 1−cos(𝛼) 1+cos(𝛼) = 1−cos(𝛼) sin(𝛼) = sin(𝛼) 1+cos(𝛼)  Double-Angle Formulas • sin(2𝜃) = 2 sin(𝜃) cos(𝜃) • cos(2𝜃) = cos2(𝜃) − sin2(𝜃) = 2cos2(𝜃) − 1 = 1 − 2sin2(𝜃) • tan(2𝜃) = 2 tan(𝜃) 1−tan2(𝜃) • sin2(𝜃) = 1−cos(2𝜃) 2 , cos2(𝜃) = 1+cos(2𝜃) 2 , tan2(𝜃) = 1−cos(2𝜃) 1−cos(2𝜃)  Product to Sum Formulas • sin(𝛼) sin(𝛽) = 1 2 [cos(𝛼 − 𝛽) − cos(𝛼 + 𝛽)] • cos(𝛼) cos(𝛽) = 1 2 [cos(𝛼 − 𝛽) + cos(𝛼 + 𝛽)] • sin(𝛼) cos(𝛽) = 1 2 [sin(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]  Sum to Product Formulas • sin(𝛼) + sin(𝛽) = 2 sin ( 𝛼+𝛽 2 ) cos ( 𝛼−𝛽 2 ) • sin(𝛼) − sin(𝛽) = 2 sin ( 𝛼−𝛽 2 ) cos ( 𝛼+𝛽 2 ) • cos(𝛼) + cos(𝛽) = 2 cos ( 𝛼+𝛽 2 ) cos ( 𝛼−𝛽 2 ) • cos(𝛼) − cos(𝛽) = −2 sin ( 𝛼+𝛽 2 ) sin ( 𝛼−𝛽 2 ) CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2018Fall  Law of Sines • sin(𝐴) 𝑎 = sin(𝐵) 𝑏 = sin(𝐶) 𝑐  Law of Cosines • 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos(𝐴) • 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos(𝐵) • 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos(𝐶)  Area of SSS Triangles (Heron’s Formula) • 𝐾 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) , where 𝑠 = 1 2 (𝑎 + 𝑏 + 𝑐)  Area of SAS Triangles • 𝐾 = 1 2 𝑎𝑏 sin (𝐶) , 𝐾 = 1 2 𝑏𝑐 sin (𝐴) , 𝐾 = 1 2 𝑎𝑐 sin (𝐵)  For 𝑦 = 𝐴sin (𝜔𝑥 − 𝜑) or 𝑦 = 𝐴cos (𝜔𝑥 − 𝜑) , with 𝜔 > 0 • Amplitude = |𝐴| , Period= 𝑇 = 2𝜋 𝜔 , Phase shift = 𝜑 𝜔