A RetroSearch Logo

Home - News ( United States | United Kingdom | Italy | Germany ) - Football scores

Search Query:

Showing content from https://dlmf.nist.gov/4.21 below:

§4.21 Identities ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.21 Identities Contents
  1. §4.21(i) Addition Formulas
  2. §4.21(ii) Squares and Products
  3. §4.21(iii) Multiples of the Argument
  4. §4.21(iv) Real and Imaginary Parts; Moduli
§4.21(i) Addition Formulas 4.21.1_5 A ⁢ cos ⁡ u + B ⁢ sin ⁡ u = A2 + B2 ⁢ cos ⁡ ( u − ph ⁡ ( A + B ⁢ i ) ) , A , B ∈ ℝ , 4.21.2 sin ⁡ ( u ± v ) = sin ⁡ u ⁢ cos ⁡ v ± cos ⁡ u ⁢ sin ⁡ v , 4.21.3 cos ⁡ ( u ± v ) = cos ⁡ u ⁢ cos ⁡ v ∓ sin ⁡ u ⁢ sin ⁡ v , 4.21.4 tan ⁡ ( u ± v ) = tan ⁡ u ± tan ⁡ v 1 ∓ tan ⁡ u ⁢ tan ⁡ v , 4.21.5 cot ⁡ ( u ± v ) = ± cot ⁡ u ⁢ cot ⁡ v − 1 cot ⁡ u ± cot ⁡ v . 4.21.6 sin ⁡ u + sin ⁡ v = 2 ⁢ sin ⁡ ( u + v 2 ) ⁢ cos ⁡ ( u − v 2 ) , 4.21.7 sin ⁡ u − sin ⁡ v = 2 ⁢ cos ⁡ ( u + v 2 ) ⁢ sin ⁡ ( u − v 2 ) , 4.21.8 cos ⁡ u + cos ⁡ v = 2 ⁢ cos ⁡ ( u + v 2 ) ⁢ cos ⁡ ( u − v 2 ) , 4.21.9 cos ⁡ u − cos ⁡ v = − 2 ⁢ sin ⁡ ( u + v 2 ) ⁢ sin ⁡ ( u − v 2 ) . §4.21(ii) Squares and Products 4.21.18 sin2 ⁡ u − sin2 ⁡ v = sin ⁡ ( u + v ) ⁢ sin ⁡ ( u − v ) , 4.21.19 cos2 ⁡ u − cos2 ⁡ v = − sin ⁡ ( u + v ) ⁢ sin ⁡ ( u − v ) , 4.21.20 cos2 ⁡ u − sin2 ⁡ v = cos ⁡ ( u + v ) ⁢ cos ⁡ ( u − v ) . §4.21(iii) Multiples of the Argument

In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities.

4.21.24 sin ⁡ ( − z ) = − sin ⁡ z , 4.21.25 cos ⁡ ( − z ) = cos ⁡ z , 4.21.26 tan ⁡ ( − z ) = − tan ⁡ z . 4.21.30 sin ⁡ ( 3 ⁢ z ) = 3 ⁢ sin ⁡ z − 4 ⁢ sin3 ⁡ z , 4.21.31 cos ⁡ ( 3 ⁢ z ) = − 3 ⁢ cos ⁡ z + 4 ⁢ cos3 ⁡ z , 4.21.32 sin ⁡ ( 4 ⁢ z ) = 8 ⁢ cos3 ⁡ z ⁢ sin ⁡ z − 4 ⁢ cos ⁡ z ⁢ sin ⁡ z , 4.21.33 cos ⁡ ( 4 ⁢ z ) = 8 ⁢ cos4 ⁡ z − 8 ⁢ cos2 ⁡ z + 1 . De Moivre’s Theorem

When n ∈ ℤ

This result is also valid when n is fractional or complex, provided that − π ≤ ℜ ⁡ z ≤ π .

4.21.35 sin ⁡ ( n ⁢ z ) = 2 n − 1 ⁢ ∏ k = 0 n − 1 sin ⁡ ( z + k ⁢ π n ) , n = 1 , 2 , 3 , … .

If t = tan ⁡ ( 1 2 ⁢ z ) , then

4.21.36 sin ⁡ z = 2 ⁢ t 1 + t2 , cos ⁡ z = 1 − t2 1 + t2 , d z = 2 1 + t2 ⁢ d t . §4.21(iv) Real and Imaginary Parts; Moduli

With z = x + i ⁢ y

4.21.37 sin ⁡ z = sin ⁡ x ⁢ cosh ⁡ y + i ⁢ cos ⁡ x ⁢ sinh ⁡ y , 4.21.38 cos ⁡ z = cos ⁡ x ⁢ cosh ⁡ y − i ⁢ sin ⁡ x ⁢ sinh ⁡ y , 4.21.39 tan ⁡ z = sin ⁡ ( 2 ⁢ x ) + i ⁢ sinh ⁡ ( 2 ⁢ y ) cos ⁡ ( 2 ⁢ x ) + cosh ⁡ ( 2 ⁢ y ) , 4.21.40 cot ⁡ z = sin ⁡ ( 2 ⁢ x ) − i ⁢ sinh ⁡ ( 2 ⁢ y ) cosh ⁡ ( 2 ⁢ y ) − cos ⁡ ( 2 ⁢ x ) .

RetroSearch is an open source project built by @garambo | Open a GitHub Issue

Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo

HTML: 3.2 | Encoding: UTF-8 | Version: 0.7.5