Trigonometry help?

Question

Write the complex number z=-4+3i in trigonometric form (polar form) with the angle expressed in radians from 0 to 2pie. Do not round any intermediate computations, and round the values in your answer to 2 decimal places. 

the place to write the answer looks like 
z= ___(cos____ + isin____) 

please help c:

Answer 1

The complex number is z = - 4 + 3i.

Compare the above equation with x + iy

Here x = - 4 and y = 3.

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Substitute the obtained values in polar form.

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Comments

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a² + b²), a = r cos θ, and b = r sin θ, and θ = tan^{- 1}(b / a) for a > 0 or θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o for a < 0.

The complex number is z = - 4 + 3i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here a = - 4 < 0 and b = 3.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (-4)² + (3)² ]

            = √[ 16 + 9 ]

            = √[ 25 ]

            = 5.

Now find the argument θ.

Since a = - 4 < 0, use the formula θ = tan^{- 1}(b / a) + 180^o.

θ = tan^{- 1}[ 3/(- 4) ] + 180^o

θ ≅ - 36.87 + 180^o

θ ≅ 143.13^o

Note that here θ is measured in degrees.

Therefore, the polar form of - 4 + 3i is about 5[ cos(143.13^o) + i sin(143.13^o) ].

Answer 2

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a² + b²), a = r cos θ, and b = r sin θ, and θ = tan^{- 1}(b / a) for a > 0 or θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o for a < 0.

The complex number is z = - 4 + 3i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here a = - 4 < 0 and b = 3.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (-4)² + (3)² ]

            = √[ 16 + 9 ]

            = √[ 25 ]

            = 5.

Now find the argument θ.

Since a = - 4 < 0, use the formula θ = tan^{- 1}(b / a) + π.

θ = tan^{- 1}[ 3/(- 4) ] + π

θ = tan^{- 1}(- 0.75) + π

θ ≅ - 0.6435 + (3.14159)

θ ≅ 2.498

θ ≅ 2.5.

Note that here θ is measured in radians.

Therefore, the polar form of - 4 + 3i is about 5 [ cos 2.5 + i sin 2.5 ].