# How do you write the following in trigonometric form and perform the operation given #(sqrt3+i)(1+i)#?

I don't really get these questions. These are easy numbers to multiply in rectangular form -- why make it harder ?

The two factors are the two biggest cliches in trig, 30/60/90 and 45/45/90, so it's OK to just write down the trigonometric forms and multiply without too much thought, as I did above.

Let's turn each factor to trigonometric form, which is

basically polar coordinates, but still a rectangular form.

Let's remember how to turn a number into trigonometric or polar form:

We'll leave it there, but we know that equals

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To write the expression ( (\sqrt{3} + i)(1 + i) ) in trigonometric form and perform the operation, we first multiply the complex numbers together:

[ (\sqrt{3} + i)(1 + i) = \sqrt{3} + i\sqrt{3} + i + i^2 ]

Next, we simplify the expression by combining like terms and using the fact that ( i^2 = -1 ):

[ = \sqrt{3} + i\sqrt{3} + i - 1 = \sqrt{3} - 1 + i(\sqrt{3} + 1) ]

Now, to express this complex number in trigonometric form, we need to find its magnitude (or modulus) and argument (or angle). The magnitude can be calculated using the Pythagorean theorem:

[ |z| = \sqrt{(\sqrt{3} - 1)^2 + (\sqrt{3} + 1)^2} ]

[ |z| = \sqrt{3 - 2\sqrt{3} + 1 + 3 + 2\sqrt{3} + 1} ]

[ |z| = \sqrt{8} = 2\sqrt{2} ]

To find the argument, we use the arctan function:

[ \theta = \arctan\left(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\right) ]

[ \theta = \arctan\left(\frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)}\right) ]

[ \theta = \arctan\left(\frac{4 + 2\sqrt{3}}{2}\right) ]

[ \theta = \arctan\left(2 + \sqrt{3}\right) ]

Finally, we express the complex number in trigonometric form:

[ z = 2\sqrt{2} \left(\cos(\arctan(2 + \sqrt{3})) + i \sin(\arctan(2 + \sqrt{3}))\right) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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