Problem 32 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$5(\cos 3+i \sin 3)$$

Step-by-Step Solution

Answer

Question: Express the given complex expression in the form $$a + bi$$: $$5(\cos 3+i \sin 3)$$ Answer: $$-4.94996 + 0.70560i$$

Step 1: Apply Euler's formula

Euler's formula states that $$e^{i\theta} = \cos \theta + i\sin \theta$$. Our given expression is already in the form of Euler's formula, so we just need to find the corresponding $$a$$ and $$b$$ values. The given expression is: $$5(\cos 3+i \sin 3)$$

Step 2: Simplify the trigonometric functions

First, we will simplify the cosine and sine functions for the angle $$3$$. Here, $$\theta = 3$$: $$\cos 3 = -0.989992, \quad \sin 3 = 0.141120$$ Approximated values are used for clarity. Then, we substitute these values into the given expression: $$5(-0.989992 + i(0.141120))$$

Step 3: Multiply by the constant and form the complex expression

Next, we will multiply each term by the constant $$5$$: $$(-0.989992 * 5) + i(0.141120 * 5)$$ $$(-4.94996) + i(0.70560)$$

Step 4: Write the final result

The expression is now in the form $$a + bi$$. The final answer is: $$-4.94996 + 0.70560i$$

Key Concepts

Euler's Formula

Euler's Formula is a fundamental bridge between exponential and trigonometric functions. It is expressed as \(e^{i\theta} = \cos \theta + i\sin \theta\). This simple formula shows how complex exponentials relate to trigonometric functions. It is immensely useful in converting between polar and rectangular forms of complex numbers.

• \(\theta\) represents the angle in radians.
• Using this formula, any complex number can be expressed in terms of its magnitude and angle.
• It's crucial for solving problems involving complex numbers in the form \(a + bi\).

For example, in the original exercise, Euler's Formula helps us understand that \(5(\cos 3 + i \sin 3)\) can be interpreted in the \(a + bi\) form by applying this relationship. This illustrates the power of Euler's Formula in manipulating and understanding complex numbers.

Trigonometric Functions

Trigonometric functions like sine and cosine are the building blocks of Euler's Formula. They describe relationships in right triangles and are crucial in relating angles to side lengths. Sine, represented as \(\sin(\theta)\), and cosine, \(\cos(\theta)\), determine the vertical and horizontal components of a point on the unit circle.

• The unit circle helps in understanding these functions, where \(\theta\) is the angle from the positive x-axis.
• Every point on this circle can be described using sine and cosine functions.
• These functions lend themselves naturally to periodic motion and waveforms.

In our problem, the trigonometric values corresponding to the angle \(3\) radians help convert the polar expression to its rectangular form using their approximated values. They are the key to finding the real and imaginary parts of a complex number.

Complex Plane

The Complex Plane is an extension of the traditional number line used in mathematics. It represents complex numbers graphically, with the horizontal axis as the real part and the vertical axis as the imaginary part. This two-dimensional plane allows for the visualization of operations with complex numbers.

• Each point on the plane corresponds to the complex number \(a + bi\).
• The plane helps in visualizing addition, subtraction, magnitude, and angle of complex numbers.
• Polar coordinates in this plane give another way to view complex numbers, using magnitude and angle.

In our exercise, understanding this plane is crucial because we are converting from polar (magnitude and direction) to rectangular coordinates (real and imaginary parts). It helps to illustrate the movement from one form—amplitude and angle—to another form—specific coordinates on the plane.

Polar to Rectangular Conversion

Polar to Rectangular Conversion is a method of translating a complex number from polar coordinates (r, \(\theta\)) to its rectangular form (a + bi). The polar form expresses a complex number by its magnitude and direction, while the rectangular form uses its horizontal and vertical components.

• To convert, use \(a = r \cos(\theta)\) and \(b = r \sin(\theta)\).
• \(r\) is the magnitude, and \(\theta\) is the angle in radians.
• This conversion connects the abstract world of complex numbers to easily interpretable real and imaginary parts.

For the given exercise, substituting the cosine and sine of \(3\) in the expressions \(a = 5 \cos(3)\) and \(b = 5 \sin(3)\) allows for straightforward computation to find \(a = -4.94996\) and \(b = 0.70560\). This conversion is vital for computation in fields where complex numbers are used extensively, such as engineering and physics.