Learning Trigonometry: Which Expression is Equivalent to (4 + 7i)(3 + 4i)? –16 + 37i 12 – 28i 16 – 37i 37 + 16i

To find the expression equivalent to (4 + 7i)(3 + 4i), we can distribute the terms using the FOIL method:

// FOIL // Simplify // Final Answer

(4 + 7i)(3 + 4i)

= 12 + 16i + 21i + 28i^2

= 12 + 37i – 28

= -16 + 37i

Therefore, the expression equivalent to (4 + 7i)(3 + 4i) is -16 + 37i.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many practical applications in engineering, physics, and astronomy. Learning trigonometry can be challenging, but it is a useful skill for anyone interested in these fields.

Pro tip: To remember the FOIL method, consider it “First, Outer, Inner, Last.”

Understanding Complex Numbers

When dealing with complex numbers, it is important to understand how to add, subtract, multiply and divide them. In this article, we will go over the basics of complex numbers and how they can be used to simplify complex expressions.

We will be looking at the expression (4 + 7i)(3 + 4i), and discussing the different ways to solve the equation to get the equivalent expression.

Introduction to Complex Numbers

Complex numbers consist of two parts – a real part and an imaginary part. They are an extension of the real numbers and are represented as a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit.

The process of multiplying complex numbers involves distributing and simplifying, much like regular algebraic expressions. In the given expression, (4 + 7i) represents the first complex number, and (3 + 4i) represents the second complex number. To find the product of these two complex numbers, we multiply the real and imaginary parts and simplify by combining like terms.

Therefore, the correct expression equivalent to (4 + 7i)(3 + 4i) is 16 + 37i.

Basic Arithmetic operations with Complex Numbers

Complex numbers are a combination of real and imaginary numbers and can be added, subtracted, multiplied, and divided like any other algebraic expressions. To solve the expression (4+7i) * (3+4i), here are the steps to follow:

– Multiply the real coefficients: (4 * 3) = 12

– Multiply the imaginary coefficients: (7i * 4i) = 28i^2

– Simplify i^2 as -1: -28

– Multiply the real and imaginary coefficients: (4i * 3) + (7i * 4)

– Simplify the result: 12-28i+12+28i

– Combine like terms, discarding the imaginary terms: 12

Therefore, the equivalent expression is 12-28i, making it the correct answer.

Multiplication of Complex Numbers

Multiplication of complex numbers involves the distribution of each term in one complex number to every term in the other complex number, followed by simplification of the resulting expression.

For instance, to multiply (4 + 7i) and (3 + 4i), we distribute each term in (4 + 7i) to every term in (3 + 4i):

(4 + 7i)(3 + 4i) =

4(3) + 4(4i) + 7i(3) + 7i(4i)

Simplifying the resulting expression gives:

12 + 16i + 21i + 28i² =

12 + 37i – 28

Therefore, (4 + 7i)(3 + 4i) is equivalent to 12 + 37i.

Simplifying Complex Multiplication

Simplifying with complex numbers can get tricky, but there are a few steps you can take to make it easier. When multiplying two complex numbers, it’s important to remember that the product of two complex numbers will also be complex.

Let’s take a look at how to simplify complex multiplication!

Foil Method (First, Outer, Inner, Last)

The foil method, which stands for First, Outer, Inner, Last, is a useful mnemonic device to simplify complex multiplication problems like (4+7i)(3+4i) in Trigonometry.

Here’s how to use the foil method:

Step Example

First: Multiply the first terms of each binomial, in this case, 4 and 3, to get 12.

Outer: Multiply the outer terms, which are 4 and 3i, to get 12i.

Inner: Multiply the inner terms, which are 7i and 3, to get 21i.

Last: Multiply the last terms, which are 7i and 4i, to get 28i2. Note that i2 is -1.

Simplify the expression by combining like terms. 12i+21i= 33i

28i2= 28(-1)= -28

Hence, the simplified expression is 12+33i-28= -16+37i.

Simplifying like terms and distributing

Simplifying like terms and distributing is crucial in solving complex multiplication problems, especially in Trigonometry.

Let’s take the expression (4 + 7i)(3 + 4i) as an example and simplify it:

Use the FOIL method (First, Outer, Inner, Last) to distribute the terms of the expression.

4 * 3 + 4 * 7i + 3 * 7i + 7i * 4i

Simplify the expression by combining like terms

12 + 28i + 28i + 28i^2

Since i^2 = -1

12 + 28i + 28i – 28

Final Answer: -16 + 37i

Following these steps and practicing such problems will help better understand and solve complex multiplication problems.

Practice Examples

To simplify complex multiplication, there are a few steps to follow to arrive at the correct answer. When solving the expression (4 + 7i)(3 + 4i), you can use the FOIL method:

First

Multiply the first terms of each binomial -> 4 x 3 = 12

Outer

Multiply the outer terms of each binomial -> 4i x 3 = 12i

Inner

Multiply the inner terms of each binomial -> 7i x 4 = 28i

Last

Multiply the last terms of each binomial -> 7i x 4i = -28

Combine like terms

12 + 12i + 28i – 28

Simplify

-16 + 40i

Therefore, the expression equivalent to (4 + 7i)(3 + 4i) is -16 + 40i.

Which Expression is Equivalent to (4 + 7i)(3 + 4i)? –16 + 37i 12 – 28i 16 – 37i 37 + 16i

When it comes to solving for the expression (4+7i)(3+4i) and finding its equivalent, it can be a bit tricky. However, you can find the answer with the right knowledge of trigonometry and proper practice. This article will discuss the steps to find the equivalent expression for (4+7i)(3+4i).

Using the Foil Method

The foil method is a straightforward technique that can multiply two binomials. To solve for (4+7i)(3+4i), follow these steps:

F – Multiply the first terms in each binomial: 4 x 3 = 12

O – Multiply the outer terms in each binomial: 4i x 3 = 12i

I – Multiply the inner terms in each binomial: 7i x 4 = 28i

L – Multiply the last terms in each binomial: 7i x 4i = 28(i^2)

Combine like terms: 12 + 12i + 28i + 28(i^2)

Simplify: 12 + 40i – 28

Final answer: -16 + 40i

Therefore, the expression that is equivalent to (4+7i)(3+4i) is -16 + 40i.

Simplifying the Multiplication

To simplify the multiplication of complex numbers such as (4+7i)(3+4i), we need to apply the FOIL rule, which stands for First, Outer, Inner, and Last.

Following the FOIL rule:

(4+7i)(3+4i) = 4 x 3 + 4 x 4i + 7i x 3 + 7i x 4i

= 12 + 16i + 21i + 28i^2

Simplifying further, 28i^2 is equal to -28:

= 12 + 37i – 28

= -16 + 37i

Therefore, the answer to (4+7i)(3+4i) is -16 + 37i.

For learning trigonometry, the expression equivalent to (4+7i)(3+4i) is 16 – 37i.

Pro Tip: Remember to simplify all parts of a complex multiplication equation by using basic algebra and the rules of exponents to arrive at the correct answer.

Checking for the Correct Answer

The correct expression equivalent to (4+7i)(3+4i) is 16+37i.

Here’s how we can solve it using trigonometry:

First, we multiply 4 and 3 (real parts) to get 12. Then, we multiply 4i and 3i (imaginary parts) to get -12. Adding these terms gives us the real part of the answer, which is 12-(-12), or 24.

Next, we multiply 4 and 4i to get 16i. We then multiply 7i and 3 to get 21i. Adding these terms gives us the imaginary part of the answer, which is 16+21, or 37.

Therefore, the answer to (4+7i)(3+4i) is 24+37i, equivalent to 16+37i when simplified.

Additional Practice Problems

Trigonometry is a form of math focusing on specific measurements such as angles, lengths, and heights. When working on a trigonometry problem, it is important to know how to simplify and combine expressions. To help you practice these skills, there are several additional practice problems that you can take on.

This section will look at one such problem and provide the answer.

Solving for (2+5i)(4+3i)

To solve for (2+5i)(4+3i), we can use the FOIL method, which stands for First, Outer, Inner, Last.

(2+5i)(4+3i) =

2*4 + 2*3i + 5i*4 + 5i*3i

= 8 + 6i + 20i – 15

= -7 + 26i

To find the expression equivalent to (4 + 7i)(3 + 4i), we use the FOIL method again:

(4+7i)(3+4i) =

4*3 + 4*4i + 7i*3 + 7i*4i

= 12 + 16i + 21i – 28

= -16 + 37i

Therefore, the expression equivalent to (4 + 7i)(3 + 4i) is -16 + 37i.

Solving for (6-2i)(1+7i)

We need to use the FOIL method to solve for (6-2i)(1+7i).

First, we multiply the first terms: 6 x 1 = 6

Then, we multiply the outside terms: 6 x 7i = 42i

Next, we multiply the inside terms: -2i x 1 = -2i

Finally, we multiply the last terms: -2i x 7i = -14i^2

Remember that i^2 = -1, so we can simplify as follows:

-14i^2 = -14(-1) = 14

Putting it all together, we get:

(6-2i)(1+7i) = 6 + 42i – 2i + 14

= 20 + 40i

To solve the additional practice problem, (4 + 7i)(3 + 4i), we follow the same FOIL method:

First, we multiply the first terms: 4 x 3 = 12

Next, we multiply the outside terms: 4 x 4i = 16i

Then, we multiply the inside terms: 7i x 3 = 21i

Finally, we multiply the last terms: 7i x 4i = -28i^2

Again, remember that i^2 = -1, so we can simplify as follows:

-28i^2 = -28(-1) = 28

Putting it all together, we get:

(4 + 7i)(3 + 4i) = 12 + 16i + 21i + 28

= 40 + 37i

Therefore, the expression equivalent to (4 + 7i)(3 + 4i) is 40 + 37i.

Pro Tip: Remember that i^2 = -1 when simplifying complex numbers.

Solving for (8-i)(4-i)

Expanding the given expression (8-i)(4-i) using the FOIL method, we get:

(8-i)(4-i) = 32 – 8i – 4i + i^2

Simplifying i^2, we get i^2 = -1

Now substituting i^2, we get:

(8-i)(4-i) = 32 – 8i – 4i – 1

= 31 -12i

Therefore, (8-i)(4-i) simplifies to 31 – 12i.

– Additional Practice Problems:

To find which expression is equivalent to (4 + 7i)(3 + 4i), we need to use the FOIL method again.

(4 + 7i)(3 + 4i) = 12 + 16i + 21i + 28i^2

Simplifying i^2, we get i^2 = -1.

Now substituting i^2, we get:

(4 + 7i)(3 + 4i) = 12 + 16i + 21i – 28

= -16 + 37i

Therefore, the correct option is -16 + 37i.

Pro Tip:  Remember to simplify i^2 to -1 while dealing with complex numbers.