Evaluate 3(cos(pi)+isin(pi))

Solution:

Evaluate the expression 3(cos(pi) + i sin(pi))

Step 1: Break down the expression

Step 1.1

Utilize the reference angle to find the equivalent trigonometric values in the first quadrant. Since the cosine is negative in the second quadrant, the expression becomes $3(-\cos (0)+i\sin (\pi ))$.

Step 1.2

The precise value of $\cos (0)$ is $1$. Thus, the expression simplifies to $3(-1\cdot 1+i\sin (\pi ))$.

Step 1.3

Carry out the multiplication of $-1$ by $1$, resulting in $3(-1+i\sin (\pi ))$.

Step 1.4

Refer to the reference angle to find the equivalent trigonometric values in the first quadrant, leading to $3(-1+i\sin (0))$.

Step 1.5

The exact value of $\sin (0)$ is $0$, simplifying the expression to $3(-1+i\cdot 0)$.

Step 1.6

Perform the multiplication of $i$ by $0$, yielding $3(-1+0)$.

Step 2: Simplify the complex expression

Step 2.1

Combine $-1$ and $0$ to get $3\cdot -1$.

Step 2.2

Multiply $3$ by $-1$ to obtain the final result, $-3$.

Knowledge Notes:

To solve this problem, we need to understand several key concepts:

1. Complex Numbers: Complex numbers are numbers in the form of $a+bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit with the property that $i^2=-1$.

2. Euler's Formula: Euler's formula states that for any real number $x$, $e^{ix}=\cos (x)+i\sin (x)$. This is particularly useful in converting from polar to rectangular form in complex numbers.

3. Trigonometric Functions: The cosine and sine functions are fundamental in trigonometry, with specific values at notable angles such as $0$, $\frac{\pi }{2}$, $\pi$, and $2\pi$. For instance, $\cos (0)=1$, $\sin (0)=0$, $\cos (\pi )=-1$, and $\sin (\pi )=0$.

4. Multiplication of Complex Numbers: When multiplying a complex number by a real number, we multiply both the real and imaginary parts by the real number.

5. Simplification: The process involves combining like terms and performing arithmetic operations to reduce the expression to its simplest form.

In the given problem, we use the known values of the cosine and sine functions at $\pi$ and $0$, and then multiply the resulting complex number by $3$ to evaluate the expression.