Convert each complex number to rectangular form. 4(cos75^∘+i sin75^∘) Hint: Use the addition f

Convert each complex number to rectangular form. 4(cos75^∘+i...

Key Concepts

Complex Numbers in Polar Form

This concept refers to expressing a complex number in the form r(cos? + i sin?), where r is the magnitude (or modulus) and ? is the argument (or angle). This form is particularly useful for multiplication, division, and understanding the geometric interpretation of complex numbers on the complex plane.

Conversion to Rectangular Form

Converting a complex number from polar form to rectangular form involves using the relationships a = r cos? and b = r sin?, where the rectangular form is written as a + bi. This conversion is essential for operations such as addition and subtraction of complex numbers, where the rectangular form is more convenient.

Trigonometric Addition Formulas

The trigonometric addition formulas, such as cos(? + ?) = cos? cos? ? sin? sin? and sin(? + ?) = sin? cos? + cos? sin?, allow one to evaluate cosine and sine for angles that are sums of simpler, well-known angles. These formulas are particularly useful in this context to compute trigonometric values for non-standard angles by expressing them as sums or differences of standard angles.

Transcript

00:01 In the given question, we are told that we have to convert each of the complex number that is given over here into rectangular form.

00:11 So the complex number that is given to us is 4 times 4 times cos of 75 degrees plus i sine of 75 degrees.

00:27 And now we are told to evaluate cos of 75 degrees and sine of 75 degrees using addition formulas.

00:37 So what we could do is we could write cos 75 over here first as cos of 45 plus 30.

00:48 And we can also write a sine 75 over here as sine of 45 plus 30.

00:56 And we are going to use the addition formulas for cos of the angles a plus b and sine of the angle a plus b.

01:09 Sine of the angle a plus b.

01:14 So when you take cos of the angle a plus b, what you get as the expansion of that is cos of a times cos of b minus sine of a times sine of b.

01:37 And similarly, if you were to take sine of a plus b, what you get is sine of a times cos of b plus sine of b times cos of a.

01:54 Now the reason why we have taken cos 45 and the angles 45 and 30 is because these are the angles we already know the values of, right? we generally use these values a lot in problems.

02:09 So we already know that cos 45 degrees and sine 45 degrees have the same value and it is equal to 1 by square root of 2.

02:21 Similarly, we know the values of cos 30 and sine 30.

02:27 Cos 30 is square root of 3 divided by 2 and sine of 30 is equal to 1 by 2.

02:36 Sine of 30, sine of 30 degrees, sine of 30 degrees is equal to 1 by 2.

02:51 So these are the values we already know that we can substitute on expanding the above formulas that we have written or expressions that we have written as cos of a plus b and cos of sine of a plus b, right? so what we could do first is we can write the expansion of cos of 45 plus 30.

03:17 It will be cos of 45 times cos of 30 minus sine of 45 times sine of 30, right? so now this would then be equal to cos 45 is 1 by square root of 2.

03:39 Cos 30 is root 3 by 2.

03:42 So we will have root 3 divided by 2 times square root of 2 minus sine 45 times sine 30 will be 1 by square root of 2 times 1 by 2, which is 1 by 2 times square root of 2.

03:59 And this we can write as, we can simplify and write as square root of 3 minus 1 divided by 2 root 2.

04:08 Going on with sine of 75, we will have sine of sine of, let's write it with a different color, we'll have sine of 45 degrees times cos of 30 degrees plus sine of 30 degrees times cos of 45 degrees...

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