07. 삼각함수의 덧셈정리(Addition and Subtraction Formulas for sine and cosine and tangent)

 

7. 1. 삼각함수의 덧셈정리(Addition and Subtraction Formulas for sine and cosine and tangent)

 

1) sin(a + b) = sin a cos b + cos a sin b.

 

    sin(a - b) = sin a cos b - cos a sin b.

 

2) cos(a + b) = cosa cos b – sin a sin b.

 

    cos(a - b) = cosa cos b + sin a sin b.

 

3) tan(a + b) = (tan a + tan b) / (1 – tan a tan b).

 

    tan(a - b) = (tan a - tan b) / (1 + tan a tan b).

 

 

- 배각의 공식.

 

  1) sin 2a = 2sin a cos a.

 

  2) cos 2a = cos² a – sin² a.

 

  3) tan 2a = 2tan a / (1 – tan² a).

 

 

- 반각의 공식.

 

  1) sin² a/2 = (1 – cos a) / 2.

 

  2) cos² a/2 = (1 + cos a) / 2.

 

  3) tan² a/2 = (1 – cos a) / (1 + cos a).

 

  4) sin² a = (1 – cos 2a) / 2.

 

  5) cos² a = (1 + cos 2a) / 2.

 

 

-  X  ->  +, -

 

  1) sin a cos b = 1/2{sin(a + b) + sin(a - b)}.

 

  2) cos a sin b = 1/2{sin(a + b) - sin(a - b)}.

 

  3) cos a cos b = 1/2{cos(a + b) + cos(a - b)}.

 

  4) sin a sin b = - 1/2{cos(a + b) + cos(a - b)}.

 

 

- +, -  ->  X

 

  1) sin A + sin B = 2sin(A + B / 2)cos(A – B / 2).

 

  2) sin A - sin B = 2cos(A + B / 2)sin(A – B / 2).

 

  3) cos A + cos B = 2cos(A + B / 2)cos(A – B / 2).

 

  4) cos A - cos B = -2sin(A + B / 2)sin(A – B / 2).

 

---

 

7. 2. 삼각함수의 합성 (Composition of Trigonometric Functions)

 

 

---

 

7. 3. 삼각함수의 극한 [x is radian] (Limits of Trigonometric Functions)

 

- lim (x → a) sin x = sin a.

 

- lim (x → a) cos x = cos a.

 

- lim (x → a) tan x = tan a. [a ≠ nπ + π/2 , n is integer]

 

 

 

- lim (x → 0) (sin x / x) = 1 , lim (x → 0) (sin bx / ax) = b/a.

 

- lim (x → 0) (tan x / x) = 1 , lim (x → 0) (tan bx / ax) = b/a.

 

- lim (x → 0) (sin x / tan x) = 1 , lim (x → 0) (sin bx / tan ax) = b/a.

 

---

 

7. 4. 삼각함수의 미분(Differentiation of Trigonometric Functions)

 

- (sin x)’ = cos x.

 

- (cos x)’ = - sin x.

 

---