Question

For questions 16 - 19, write each expression in the standard form for the complex number a + bi. ( 2 points each)

Answer 1

Standard form for the complex number is
`\[\left(16\cos^2 \left((7 x)/(9)\right)+112 \sin \left((7 \pi)/(9)\right)\cos \left((7 x)/(9)\right)-49 \sin^2 \left((7 \pi)/(9)\right)\right)+0i\]\\`

To expand the square, we need to multiply the expression by itself. So, we have:

`\[4\left(\cos \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\right) \cdot 4\left(\cos \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\right)\]`

We distribute each term in the first expression to each term in the second expression:

`\[16\left(\cos \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\right)\left(\cos \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\right)\]`

Using the FOIL method (First, Outer, Inner, Last), we multiply the terms:

`\[16\left(\cos^2 \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\cos \left((7 x)/(9)\right)+7 \sin \left((7 \pi)/(9)\right)\cos \left((7 x)/(9)\right)+49 \sin^2 \left((7 \pi)/(9)\right)\right)\]`

Simplify the expression by combining like terms:

`\[16\left(\cos^2 \left((7 x)/(9)\right)+14 \sin \left((7 \pi)/(9)\right)\cos \left((7 x)/(9)\right)+49 \sin^2 \left((7 \pi)/(9)\right)\right)\]`

To write the expression in standard form a+bi, we need to separate the real part (a) and the imaginary part (b).

The real part will consist of the terms with cosines, and the imaginary part will consist of the terms with sines. So, we have:

`\[\left(16\cos^2 \left((7 x)/(9)\right)+112 \sin \left((7 \pi)/(9)\right)\cos \left((7 x)/(9)\right)-49 \sin^2 \left((7 \pi)/(9)\right)\right)+0i\]`

This is the standard form of the complex number.