Exam Notes

Equations

  • Pythagorean Theorem: a2+b2=c2a^2 + b^2 = c^2, example: 62−52=c262 - 5^2 = c^2
  • Sum of angles: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A \cos B + \cos A \sin B and cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A \cos B - \sin A \sin B

Binomial Expansion

  • Coefficients (rows of Pascal's triangle): 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1.
  • Example: (2x+y)3(2x + y)^3

Law of Sines and Cosines

  • Law of Sines: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
  • Law of Cosines: Used when given A+B.

Vectors

  • Component of a vector: If (1,2)(1, 2) and (5,7)(5, 7), then component is (4,5)(4, 5).
  • Magnitude: 82+62\sqrt{8^2 + 6^2}

Area Formulas

  • Area of a triangle: A=12bh=12bcsin⁡A=12absin⁡C=12acsin⁡BA = \frac{1}{2}bh = \frac{1}{2}bc \sin A = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B
  • Heron's formula: A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)} where s=a+b+c2s = \frac{a+b+c}{2}.

Other

  • Binomial coefficient: (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}
  • Dot product: For vectors <4,2><4, 2> and <3,1><3, 1>, the dot product is (4<em>3)+(2</em>1)=14(4<em>3) + (2</em>1) = 14
  • Trigonometry: sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1