UNIT 1
수열
\(a_n = a_1 + (n-1)d\) | \(S_n = \dfrac{n}{2}(2a_1+(n-1)d)\)
\(a_n = a_1 r^{n-1}\) | \(S_n = \dfrac{a_1(r^n-1)}{r-1} \,(r\ne1)\)
\(\sum k\)\(\dfrac{n(n+1)}{2}\)
\(\sum k^2\)\(\dfrac{n(n+1)(2n+1)}{6}\)
\(\sum k^3\)\(\left[\dfrac{n(n+1)}{2}\right]^2\)
부분분수\(\dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}\)
📝 예제
\(a_3=7,\; a_7=19\)인 등차수열에서 \(S_8\)을 구하여라.
공차 \(d=3\), \(a_1=1\). \(S_8=\dfrac{8}{2}(2+7\cdot3)=4\times23=\mathbf{92}\)
UNIT 2
지수 · 로그
\(a^m\cdot a^n=a^{m+n},\quad (a^m)^n=a^{mn},\quad (ab)^n=a^n b^n\)
\(\log_a MN=\log_a M+\log_a N,\quad \log_a M^k=k\log_a M\)
밑 변환: \(\log_a b=\dfrac{\log_c b}{\log_c a}\)
지수함수\(y=a^x\,(a>0,a\ne1)\)
로그함수정의역 \(x>0\)
\(\log 2\)\(\approx 0.3010\)
\(\log 3\)\(\approx 0.4771\)
📝 예제
\(\log_2 3=a\)일 때 \(\log_4 12\)를 \(a\)로 나타내어라.
\(\log_4 12=\dfrac{\log_2 12}{\log_2 4}=\dfrac{\log_2(4\cdot3)}{2}=\dfrac{2+a}{2}\)
UNIT 3
삼각함수
\(\sin^2\theta+\cos^2\theta=1,\quad \tan\theta=\dfrac{\sin\theta}{\cos\theta}\)
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)
\(\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\)
사인법칙: \(\dfrac{a}{\sin A}=2R\) 코사인법칙: \(a^2=b^2+c^2-2bc\cos A\)
sin 30°\(\tfrac{1}{2}\)
cos 30°\(\tfrac{\sqrt3}{2}\)
sin 45°\(\tfrac{\sqrt2}{2}\)
tan 60°\(\sqrt3\)
📝 예제
\(\sin\theta+\cos\theta=\sqrt2\)일 때 \(\sin\theta\cos\theta\)의 값은?
양변 제곱: \(1+2\sin\theta\cos\theta=2\) → \(\sin\theta\cos\theta=\dfrac{1}{2}\)