Complex Numbers Z1, Z2, Z3 And Trigonometric Values Cos(5π/12), Sin(5π/12)

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In the world of mathematics, complex numbers extend the real number system by including the imaginary unit 'i', defined as the square root of -1. These numbers, expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, open up a vast realm of mathematical possibilities. This article delves into the intricacies of complex numbers, exploring their algebraic and trigonometric forms, and demonstrating how they can be used to deduce trigonometric values.

Part (a): Unveiling z3 in Algebraic Form

In this section, we are given three complex numbers: z1 = 1 + i, z2 = -1 - 3i, and z3 = 1 / z2. Our mission is to express z3 in its algebraic form, which is a + bi, where 'a' and 'b' are real numbers. To achieve this, we will first determine the value of z3 by taking the reciprocal of z2. Subsequently, we will rationalize the denominator of the resulting complex fraction to arrive at the desired algebraic form. The given value of z2 is -1 - 3i. To find z3, which is the reciprocal of z2, we calculate 1 / (-1 - 3i). This fraction has a complex number in the denominator, which we need to rationalize to express it in the standard algebraic form. Rationalizing the denominator involves multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of -1 - 3i is -1 + 3i. By multiplying the numerator and denominator by this conjugate, we eliminate the imaginary part from the denominator. Now, let’s perform the multiplication: (1 / (-1 - 3i)) * ((-1 + 3i) / (-1 + 3i)). Multiplying the numerators gives us -1 + 3i. For the denominators, we use the formula (a - b)(a + b) = a^2 - b^2, where a = -1 and b = 3i. Thus, the denominator becomes (-1)^2 - (3i)^2 = 1 - 9i^2. Since i^2 = -1, the denominator simplifies to 1 - 9(-1) = 1 + 9 = 10. Now, we have z3 = (-1 + 3i) / 10. To express z3 in the algebraic form a + bi, we separate the real and imaginary parts: z3 = -1/10 + (3/10)i. Therefore, z3 in algebraic form is -0.1 + 0.3i. This result provides a clear representation of z3 as a complex number with a real part of -0.1 and an imaginary part of 0.3.

Part (b): Expressing z3 in Trigonometric Form

After successfully expressing z3 in its algebraic form, we now embark on the journey of transforming it into its trigonometric form. The trigonometric form of a complex number provides an alternative representation that highlights its magnitude and direction in the complex plane. This form is particularly useful in various mathematical and engineering applications, offering a different perspective on complex number operations. The trigonometric form of a complex number z is given by r(cos θ + i sin θ), where r is the modulus (or magnitude) of z, and θ is the argument (or angle) of z. To convert z3 = -0.1 + 0.3i into trigonometric form, we first need to calculate its modulus, denoted as |z3|. The modulus is the distance from the origin to the point representing z3 in the complex plane and is calculated using the formula |z3| = √(a^2 + b^2), where a is the real part and b is the imaginary part of z3. In this case, a = -0.1 and b = 0.3. Substituting these values into the formula, we get |z3| = √((-0.1)^2 + (0.3)^2) = √(0.01 + 0.09) = √0.10. Thus, the modulus of z3 is √0.10, which can also be expressed as √(1/10) or 1/√10. Next, we need to find the argument θ of z3. The argument is the angle between the positive real axis and the line connecting the origin to the point representing z3 in the complex plane. The argument can be found using the arctangent function, θ = arctan(b/a). However, it is crucial to consider the quadrant in which z3 lies to determine the correct angle. Since z3 = -0.1 + 0.3i has a negative real part and a positive imaginary part, it lies in the second quadrant of the complex plane. The arctangent function will give us an angle in the range (-π/2, π/2), so we need to adjust it to the correct quadrant. First, let's calculate the reference angle α = arctan(|b/a|) = arctan(|0.3 / -0.1|) = arctan(3). Using a calculator, we find that arctan(3) ≈ 1.249 radians. Since z3 is in the second quadrant, the argument θ is given by θ = π - α. Therefore, θ ≈ π - 1.249 ≈ 1.893 radians. Now that we have the modulus |z3| = 1/√10 and the argument θ ≈ 1.893 radians, we can express z3 in trigonometric form: z3 = (1/√10)(cos(1.893) + i sin(1.893)). This trigonometric form provides a clear visualization of z3 in the complex plane, highlighting its magnitude and direction. The modulus represents the length of the vector from the origin to z3, and the argument represents the angle this vector makes with the positive real axis.

Part (c): Deducing Values of cos(5π/12) and sin(5π/12)

This final part of our exploration delves into the fascinating application of complex numbers in deducing trigonometric values. We will leverage the results obtained in the previous sections, specifically the algebraic and trigonometric forms of z3, to extract the values of cos(5π/12) and sin(5π/12). This demonstration highlights the powerful connection between complex numbers and trigonometry, showcasing how one can be used to illuminate the other. The strategy we will employ involves equating the algebraic and trigonometric forms of z3 and then comparing their real and imaginary parts. This comparison will yield two equations, which we can then solve to determine the values of cos(5π/12) and sin(5π/12). From Part (a), we know that z3 in algebraic form is -0.1 + 0.3i, which can also be written as -1/10 + (3/10)i. From Part (b), we have z3 in trigonometric form as (1/√10)(cos(1.893) + i sin(1.893)). However, to directly deduce the values of cos(5π/12) and sin(5π/12), we need to express the argument 1.893 radians in terms of π. We know that 5π/12 radians is equal to 5 * 180 / 12 = 75 degrees. Let’s convert this to radians: 75 degrees * (π / 180) = 5π/12 radians. Therefore, we aim to find cos(5π/12) and sin(5π/12). Now, let’s rewrite the trigonometric form of z3 using the argument 5π/12: z3 = (1/√10)(cos(5π/12) + i sin(5π/12)). Equating the algebraic and trigonometric forms, we have: -1/10 + (3/10)i = (1/√10)(cos(5π/12) + i sin(5π/12)). To proceed, we multiply both sides by √10: √10 * (-1/10 + (3/10)i) = cos(5π/12) + i sin(5π/12). This simplifies to: -√10/10 + (3√10/10)i = cos(5π/12) + i sin(5π/12). Now, we can equate the real and imaginary parts: cos(5π/12) = -√10/10 sin(5π/12) = 3√10/10. Thus, we have deduced the values of cos(5π/12) and sin(5π/12) using complex numbers. The cosine of 5π/12 is -√10/10, and the sine of 5π/12 is 3√10/10. This elegant demonstration showcases the interplay between complex numbers and trigonometry, providing a powerful tool for solving trigonometric problems. By leveraging the algebraic and trigonometric forms of complex numbers, we can unveil hidden relationships and extract valuable information.

In conclusion, this exploration has navigated the realm of complex numbers, revealing their multifaceted nature and their connection to trigonometry. We have successfully expressed z3 in both algebraic and trigonometric forms, and we have harnessed these representations to deduce the values of cos(5π/12) and sin(5π/12). This journey underscores the beauty and power of mathematics, where seemingly disparate concepts intertwine to illuminate deeper truths.