Simplify Completely Into An Expression With $\sin(A$\] Or $\cos(A$\] Only:$1 - (\sin(A) + \cos(A))^2 = \square$

Feb 28, 2025 by ADMIN 112 views

Simplify Completely into an Expression with $\sin(A)$ or $\cos(A)$ Only: $1 - (\sin(A) + \cos(A))^2 = \square$

In mathematics, trigonometric identities are essential for solving problems involving triangles and waves. One of the fundamental identities is the Pythagorean identity, which states that $\sin^2(A) + \cos^2(A) = 1$ . However, in this article, we will focus on simplifying an expression involving the sum of sine and cosine functions. The given expression is $1 - (\sin(A) + \cos(A))^2$ , and we will simplify it into an expression containing only $\sin(A)$ or $\cos(A)$ .

The given expression is $1 - (\sin(A) + \cos(A))^2$ . To simplify this expression, we need to expand the squared term using the formula $(a + b)^2 = a^2 + 2ab + b^2$ . Applying this formula, we get:

1 - (\sin(A) + \cos(A))^2 = 1 - (\sin^2(A) + 2\sin(A)\cos(A) + \cos^2(A))

Now, let's expand the expression further by distributing the negative sign to each term inside the parentheses:

1 - \sin^2(A) - 2\sin(A)\cos(A) - \cos^2(A)

We can simplify the expression by combining like terms. The first and last terms are $1$ and $-\cos^2(A)$ , respectively. The middle terms are $-\sin^2(A)$ and $-2\sin(A)\cos(A)$ . Combining these terms, we get:

1 - \cos^2(A) - \sin^2(A) - 2\sin(A)\cos(A)

Now, let's apply the Pythagorean identity, which states that $\sin^2(A) + \cos^2(A) = 1$ . We can rewrite the expression as:

1 - \cos^2(A) - \sin^2(A) - 2\sin(A)\cos(A) = 1 - (\sin^2(A) + \cos^2(A)) - 2\sin(A)\cos(A)

Using the Pythagorean identity, we can simplify the expression further:

1 - (\sin^2(A) + \cos^2(A)) - 2\sin(A)\cos(A) = 1 - 1 - 2\sin(A)\cos(A)

Now, let's simplify the expression further by combining the constants:

1 - 1 - 2\sin(A)\cos(A) = -2\sin(A)\cos(A)

In this article, we simplified the expression $1 - (\sin(A) + \cos(A))^2$ into an expression containing only $\sin(A)$ or $\cos(A)$ . The final simplified expression is $-2\sin(A)\cos(A)$ . This expression can be used to solve problems involving the sum of sine and cosine functions.

Here are some additional examples of simplifying expressions involving the sum of sine and cosine functions:

$1 - (\sin(A) - \cos(A))^2 = 1 - (\sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A))$
$1 - (\sin(A) + 2\cos(A))^2 = 1 - (\sin^2(A) + 4\sin(A)\cos(A) + 4\cos^2(A))$

Here are some tips and tricks for simplifying expressions involving the sum of sine and cosine functions:

Use the Pythagorean identity to simplify expressions involving $\sin^2(A)$ and $\cos^2(A)$ .
Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ to expand squared terms.
Combine like terms to simplify expressions.

Here are some common mistakes to avoid when simplifying expressions involving the sum of sine and cosine functions:

Failing to apply the Pythagorean identity.
Failing to expand squared terms using the formula $(a + b)^2 = a^2 + 2ab + b^2$ .
Failing to combine like terms.

In conclusion, simplifying expressions involving the sum of sine and cosine functions requires careful application of trigonometric identities and formulas. By following the tips and tricks outlined in this article, you can simplify expressions involving the sum of sine and cosine functions and solve problems involving triangles and waves.
Simplify Completely into an Expression with $\sin(A)$ or $\cos(A)$ Only: $1 - (\sin(A) + \cos(A))^2 = \square$ - Q&A

In our previous article, we simplified the expression $1 - (\sin(A) + \cos(A))^2$ into an expression containing only $\sin(A)$ or $\cos(A)$ . However, we received many questions from readers regarding the simplification process. In this article, we will address some of the most frequently asked questions and provide additional examples to help you understand the concept better.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental trigonometric identity that states $\sin^2(A) + \cos^2(A) = 1$ . This identity is used to simplify expressions involving $\sin^2(A)$ and $\cos^2(A)$ .

Q: How do I apply the Pythagorean identity?

A: To apply the Pythagorean identity, simply substitute $\sin^2(A) + \cos^2(A)$ with $1$ in the expression. For example, if you have the expression $1 - \sin^2(A) - \cos^2(A)$ , you can simplify it to $1 - (1 - \sin^2(A) - \cos^2(A)) = \sin^2(A) + \cos^2(A)$ .

Q: What is the formula for expanding squared terms?

A: The formula for expanding squared terms is $(a + b)^2 = a^2 + 2ab + b^2$ . This formula is used to expand expressions involving squared terms.

Q: How do I simplify expressions involving the sum of sine and cosine functions?

A: To simplify expressions involving the sum of sine and cosine functions, follow these steps:

Expand the squared term using the formula $(a + b)^2 = a^2 + 2ab + b^2$ .
Apply the Pythagorean identity to simplify expressions involving $\sin^2(A)$ and $\cos^2(A)$ .
Combine like terms to simplify the expression.

Q: What are some common mistakes to avoid when simplifying expressions involving the sum of sine and cosine functions?

A: Some common mistakes to avoid when simplifying expressions involving the sum of sine and cosine functions include:

Failing to apply the Pythagorean identity.
Failing to expand squared terms using the formula $(a + b)^2 = a^2 + 2ab + b^2$ .
Failing to combine like terms.

Q: Can you provide additional examples of simplifying expressions involving the sum of sine and cosine functions?

A: Here are some additional examples:

$1 - (\sin(A) - \cos(A))^2 = 1 - (\sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A))$
$1 - (\sin(A) + 2\cos(A))^2 = 1 - (\sin^2(A) + 4\sin(A)\cos(A) + 4\cos^2(A))$

Here are some additional resources to help you learn more about simplifying expressions involving the sum of sine and cosine functions:

Khan Academy: Trigonometry
MIT OpenCourseWare: Trigonometry
Wolfram Alpha: Trigonometry

Here are some practice problems to help you practice simplifying expressions involving the sum of sine and cosine functions:

Simplify the expression $1 - (\sin(A) + \cos(A))^2$ .
Simplify the expression $1 - (\sin(A) - \cos(A))^2$ .
Simplify the expression $1 - (\sin(A) + 2\cos(A))^2$ .

Here are the answers to the practice problems:

$1 - (\sin(A) + \cos(A))^2 = -2\sin(A)\cos(A)$
$1 - (\sin(A) - \cos(A))^2 = -2\sin(A)\cos(A)$
$1 - (\sin(A) + 2\cos(A))^2 = -4\sin(A)\cos(A) - 3$