#### Completing sine implementation No. 120

##### Transforming arguments. Q: We've reached an implementation offering good results for $sin ⁡ ( x )$ if $x ∈ [ - π 2 , π 2 ]$. The following rules may be used to retain precision for arbitrary argument values: $∀ x ∈ ℝ , ∀ n ∈ ℤ : sin ⁡ ( 2 ⁢ π ⁢ n + x ) = sin ⁡ ( x )$ This rule of periodicity allows us to consider only the interval $[ - π , π [$ like e.g. $sin ⁡ ( 21 ) = sin ⁡ ( 21 - 3 × 2 ⁢ π ) ≈ sin ⁡ ( 2.15 )$. $∀ x ∈ ℝ : sin ⁡ ( x ) = sin ⁡ ( π - x )$ This rule allows us to narrow down values from $[ - π , π [$ to $[ - π 2 , π 2 [$ like e.g. $sin ⁡ (2 ) = sin ⁡ ( π - 2 ) ≈ sin ⁡ ( 1.14 )$. $∀ x ∈ ℝ : sin ⁡ ( x ) = cos ⁡ ( π 2 - x )$ This rule allows us to further narrow down values from $[ - π 2 , π 2 [$ to $[ - π 4 , π 4 [$ like e.g. $sin ⁡ (1.1 ) = cos ⁡ ( π 2 - 1.1 ) ≈ cos ⁡ ( 0.471 )$. We must however implement the corresponding power series of $cos ⁡ ( x )$: Equation 4. Power series definition of $cos ⁡ ( x )$ $cos ⁡ ( x ) = 1 - x 2 2 ! + x 4 4 ! - x 6 6 ! + ... = ∑ k = 0 ∞ ( -1 ) k x 2 k ( 2 k ) !$ The above rules allow for computation of arbitrary $sin ⁡ ( x )$ values by means of power series expansion limited to the interval $[ - π 4 , π 4 ]$ thereby gaining high precision results. Extend your current implementation by mapping arbitrary arguments to this interval appropriately. Hint: The standard function Math.rint(double) could be helpful: It may turn e.g. 4.47 (double) to 4 (long). A: Maven module source code available at sub directory P/Sd1/math/V4 below lecture notes' source code root, see hints regarding import. Online browsing of API and implementation. For convenience reasons we start defining PI within our class: public class Math { private static final double PI = java.lang.Math.PI; ...Now we need two steps mapping our argument:public static double sin(double x) { // Step 1: Normalize x to [-PI, +PI[ final long countTimes2PI = (long) java.lang.Math.rint(x / 2 / PI); if (countTimes2PI != 0) { x -= 2 * PI * countTimes2PI; } // Step 2: Normalize x to [-PI/2, +PI/2] // Since sin(x) = sin (PI - x) we continue to normalize // having x in [-PI/2, +PI/2] if (PI/2 < x) { x = PI - x; } else if (x < -PI/2) { x = -x - PI; } // Step 3: Continue with business as usual ...This yet sows only the result from applying the first two rules. You may also view the Javadoc and the implementation of double Math.sin(double).