Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\left(\cos 290^{\circ}+i \sin 290^{\circ}\right)\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)$$

Step-by-Step Solution

Answer

The result of the operation is \( -\cos50^{\circ} + i \sin50^{\circ} \).

Step 1: Identify and Write Down the Formulas

We start by identifying the main variables and the formulas to be used. The formulas for the cosine and sine of the sum of two angles are: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] and \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]. Here, \( A = 290^{\circ} \) and \( B = 200^{\circ} \).

Step 2: Multiply the Complex Numbers

Multiplying the complex numbers in the task, we'll combine the real and imaginary parts separately. We have to use the formulas identified earlier to calculate the trigonometric functions of the sum of angles. For the real part, we will have \[ \cos(290^{\circ} + 200^{\circ}) = \cos 290^{\circ} \cos 200^{\circ} - \sin 290^{\circ} \sin 200^{\circ} \], and for the imaginary part, we will have \[ \sin(290^{\circ} + 200^{\circ}) = \sin 290^{\circ} \cos 200^{\circ} + \cos 290^{\circ} \sin 200^{\circ} \].

Step 3: Simplify the Result

After substituting the values of the trigonometric functions at the required angles, we find: \[ \cos(490^{\circ}) = \cos(130^{\circ}) = -\cos(50^{\circ}) \] as cosine function has period of 360 degrees and \[ \sin(490^{\circ}) = \sin(130^{\circ}) = \sin(50^{\circ}) \] as sine function also has period of 360 degrees. Thus, the final result is: \[ -\cos50^{\circ} + i \sin50^{\circ} \].

Key Concepts

Angle Sum Formulas

When dealing with trigonometric functions such as cosine and sine, the angle sum formulas are fundamental tools. These formulas help us compute the values of the trigonometric functions for the sum of two angles.

The formulas are:

• For cosine: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• For sine: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

These formulas allow us to break down complex-looking trigonometric expressions into more manageable computations, serving as a bridge to simplify the process of multiplication and addition within the complex plane.

By using these formulas, we substitute the necessary angles and solve the expressions step by step. This technique is particularly useful in simplifying the trigonometric form of complex numbers when multiplied. Always remember to consider the periodicity of the trigonometric functions when the angle exceeds 360 degrees, as they repeat every full circle.

Complex Numbers Multiplication

Complex numbers have both a real part and an imaginary part. In trigonometric form, a complex number is expressed as \( \cos \theta + i \sin \theta \). When multiplying two complex numbers in this form, we leverage the angle sum formulas to determine the result.

The multiplication of \((\cos A + i \sin A)\) and \((\cos B + i \sin B)\) can be performed by applying the angle sum for cosine and sine:

• For the real part use: \( \cos(A + B) \)
• For the imaginary part use: \( \sin(A + B) \)

The result is also a complex number in trigonometric form, where the angle is the sum of the two initial angles.

This method of multiplying complex numbers offers an elegant approach to simplify calculations that would otherwise require more cumbersome algebraic manipulation. Transforming the product into a form that is easy to interpret can help us understand the behavior and properties of complex numbers in terms of their trigonometric representations.

Cosine and Sine Functions

Cosine and sine are vital trigonometric functions with periodic properties. Each function behaves cyclically, repeating every \(360^{\circ}\) for degrees or \(2\pi\) for radians.

In the realm of complex numbers and trigonometry, these functions reveal their importance when representing complex numbers in polar or trigonometric form. When angles surpass \(360^{\circ}\), we can reduce them to equivalent angles within the first full circle by subtracting \(360^{\circ}\).

For example, if you compute \( \cos(490^{\circ}) \), you convert it to \( \cos(130^{\circ}) \) by subtracting 360 degrees. This property allows us to find equivalent angles that are easier to work with.

• Cosine function: symmetric around \(0^{\circ}\) and \(180^{\circ}\)
• Sine function: symmetric around \(90^{\circ}\) and \(270^{\circ}\)

These symmetries help us understand why certain angles yield equivalent values and ensure that we can simplify the processes involving angles greater than the typical cycles. Thus, understanding these properties is crucial when solving complex trigonometric equations.