Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$(1+\sin t)^{2}+\cos ^{2} t$$

Step-by-Step Solution

Answer

The expression simplifies to \(2(1 + \sin t)\).

Step 1: Expand the Square

To solve the expression \((1 + \sin t)^2 + \cos^2 t\), first expand the square by applying the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 1\) and \(b = \sin t\). The expansion is: \[1^2 + 2(1)(\sin t) + (\sin t)^2 = 1 + 2\sin t + \sin^2 t\].

Step 2: Substitute Identity

In the expanded expression \(1 + 2\sin t + \sin^2 t + \cos^2 t\), use the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). Substitute \(\sin^2 t + \cos^2 t\) with 1: \[1 + 2\sin t + 1 = 2 + 2\sin t\].

Step 3: Simplify the Result

Simplify the expression \(2 + 2\sin t\) by factoring out 2: \[2(1 + \sin t)\]. The expression is now fully simplified.

Key Concepts

Simplification

Simplification is an important step in solving mathematical expressions. It involves reducing complex equations to a simpler form that is easier to work with. In the exercise given, simplification is first seen after using the expansion formula for the squared term.
By expanding \((1 + \sin t)^2\) to \(1 + 2\sin t + \sin^2 t\), we make the initial expression easier to manage. The subsequent step involves using a trigonometric identity, which further simplifies the expression to \(2 + 2\sin t\).
Through factoring, we achieve the final simplified form of the expression, \(2(1 + \sin t)\), which is the most condensed version of the given problem. Simplification helps in clearly understanding and solving mathematical problems by removing unnecessary complexity.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (+, −, ×, ÷). They can also include exponents and, like in this exercise, trigonometric functions. Understanding algebraic expressions is crucial because it allows one to perform operations like addition, subtraction, and factoring.

In our problem, the expression \((1 + \sin t)^{2} + \cos^{2} t\) is an algebraic expression involving both a squared term and a trigonometric identity.
Recognizing these elements helps in applying the appropriate identities and operations, like expansion and substitution, to simplify. Dealing with algebraic expressions often involves making strategic decisions on which rules and identities to apply, as seen in the use of \((a + b)^2 = a^2 + 2ab + b^2\), leading to the final simplified form.

Trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles and the functions derived from them like sine, cosine, and tangent. When expressions involve trigonometric functions, utilizing known identities can significantly simplify the process of solving them.

In the given exercise, the key trigonometric identity used is \(\sin^2 t + \cos^2 t = 1\). This identity is fundamental and appears frequently in many problems.
By recognizing and substituting this identity into our expression \(\sin^2 t + \cos^2 t\), we reduce the complexity of the problem. This allows us to focus on more manageable algebraic manipulations and simplify the initial trigonometric expression to reach the final answer.

Factoring

Factoring is a mathematical process used to express a polynomial as a product of its factors. It's a critical tool for simplifying expressions and solving equations. In this exercise, after expanding and simplifying, we reach \(2 + 2\sin t\).

By noticing that \(2\) is a common factor in both terms, we can factor it out, resulting in \(2(1 + \sin t)\). This step is crucial as it presents the simplest and most elegant form of the expression.
Recognizing the opportunity to factor is essential because it can make a complex expression much simpler, like turning \(2 + 2\sin t\) into \(2(1 + \sin t)\). Factoring streamlines a solution and gives clarity to the mathematical relationships within the expression.