In this problem, you will prove the sin and cos sum formulas in two ways. sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b Use Euler’s formula: eia=cos⁡a+isin⁡a to prove the formulas. Use the rotation matrix Rθ from last term and the fact that the matrix of a com- position of two linear transformations is the product of their respective matrices. (therefore RθRv=Rθ+v)

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wheezym · Accepted Answer

Step 1 We need to prove that sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b. cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b. First method using Euler&#039;s formula eia=cos⁡a+isin⁡a . Properties of exponentials ea+b=ea⋅eb. ei(a+b)=eia+ib cos⁡(a+b)+isin⁡(a+b)=eiaeib cos⁡(a+b)+isin⁡(a+b)=eiaeib =(cos⁡a+isin⁡a)(cos⁡b+isin⁡b) cos⁡(a+b)+isin⁡(a+b)=(cos⁡acos⁡b+isin⁡acos⁡b+icos⁡asin⁡b−sin⁡asin⁡b) cos⁡(a+b)+isin⁡(a+b)=((cos⁡acos⁡b−sin⁡asin⁡b)+i(sin⁡acos⁡b+cos⁡asin⁡b)) Two complex numbers are equal if and only if their real and imaginary parts are equal. Therefore,cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b This completes the proof. Step 2 We will prove the same using rotation of matrix . Composition of two linear transformation is product of their respective matrices. Rθ=[cos⁡θ−sin⁡θsinθcosθ]. Form the properties of rotation matrices we know that Rθ+v=RθRv.

In this problem, you will prove the sin and cos sum formulas in two ways.PS

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