Question

Use De Moivre's theorem to express cos3a in terms of

cosa and
sin3a in terms of
sina.

Answer 1

To express cos3a in terms of cosa and sin3a in terms of sina using De Moivre's theorem, we can use the binomial expansion and simplify the expressions.

Step-by-step explanation:

To express cos3a in terms of cosa using De Moivre's theorem, we can use the formula: cos(nx) = Re[(cos(x) + i * sin(x))n].

In this case, n = 3 and x = a, so we have cos(3a) = Re[(cos(a) + i * sin(a))3].

Using binomial expansion, we can expand (cos(a) + i * sin(a))3 to get:

cos(3a) = Re[cos(a)3 + 3 * cos(a)2 * i * sin(a) + 3 * cos(a) * i2 * sin(a)2 + i3 * sin(a)3].

Since i2 = -1 and i3 = -i, we can simplify this expression:

cos(3a) = cos(a)3 - 3 * cos(a) * sin(a)2.

Similarly, to express sin3a in terms of sina, we can use the same formula and simplify to get:

sin(3a) = 3 * cos(a)2 * sin(a) - sin(a)3.