Problem 5 - Precalculus

Question

Carry out the indicated operations. (a) \((1+T)^{2}\) (b) \((1+\tan \theta)^{2}\)

Step-by-Step Solution

Answer

(a) \(1 + 2T + T^2\); (b) \(1 + 2\tan \theta + \tan^2 \theta\)."

Step 1: Expand the binomial square formula for part (a)

To solve \((1+T)^{2}\), apply the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a=1\) and \(b=T\). Substitute these into the formula: \((1+T)^2 = 1^2 + 2 \cdot 1 \cdot T + T^2\). Simplify it to get \(1 + 2T + T^2\).

Step 2: Expand the binomial square for part (b)

Similarly, expand \((1 + \tan \theta)^{2}\) using the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a=1\) and \(b=\tan \theta\). Thus, \((1 + \tan \theta)^2 = 1^2 + 2 \cdot 1 \cdot \tan \theta + (\tan \theta)^2\). After simplification, this becomes \(1 + 2\tan \theta + \tan^2 \theta\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, consisting of numbers, variables, and operators. They allow us to perform arithmetic operations within the rules of algebra to solve complex problems. In this exercise, we encounter expressions like \((1+T)^{2}\) and \((1+\tan \theta)^{2}\), which consist of constants and variables raised to a power.

Algebraic expressions can often be simplified or expanded using various techniques. Applying these techniques makes it easier to work with and understand them.

• **Constants:** Fixed numbers like 1 in the expressions.
• **Variables:** Symbols like \(T\) and \(\theta\) that can represent numbers.
• **Operations:** Inclusion of addition, subtraction, multiplication, and division.

Understanding how to manipulate algebraic expressions is crucial for solving mathematical problems, from basic equations to complex algebraic structures.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of terms. The Binomial Theorem is commonly used for expanding polynomials in the form of \((a+b)^n\). For our exercises, we used the specific form of the Binomial Theorem for squares:
\[(a+b)^2 = a^2 + 2ab + b^2\]
This formula helps in converting the square of a binomial into a simple polynomial. Let's break it down further:

• **Square of the first term:** In \((1+T)^2\), \(1^2 = 1\), and similarly for \((1+\tan \theta)^2\), \(1^2 = 1\).
• **Twice the product of the terms:** For \((1+T)^2\), twice the product is \(2 \cdot 1 \cdot T = 2T\); for \((1+\tan \theta)^2\), it is \(2 \cdot 1 \cdot \tan \theta = 2\tan \theta\).
• **Square of the second term:** The square of \(T\) gives \(T^2\), and that of \(\tan \theta\) is \((\tan \theta)^2\).

By expanding binomials using this method, we simplify complex expressions, making them easier to evaluate or further manipulate.

Trigonometric Functions

Trigonometric functions like tangent (\(\tan\)) are foundational in mathematics, especially in problems combining algebra and geometry. They relate angles of triangles to the ratios of the sides. When integrated into algebraic expressions, they require careful treatment, especially during operations like expansion or simplification.

In the expression \((1+\tan \theta)^2\), \(\tan \theta\) operates as a variable whose value depends on angle \(\theta\). The expansion *\(1 + 2\tan \theta + \tan^2 \theta\)* captures how \(\tan \theta\) interacts with other terms:

• **\(\tan \theta \)** represents the tangent of an angle \(\theta\), often denoted as the ratio of opposite to adjacent sides in a right triangle.
• **Squaring \(\tan \theta \)** signifies multiplying the function by itself, resulting in \((\tan \theta)^2\), which is essential in many trigonometric identities.

Trigonometric functions in algebraic expressions expand not just within geometry, but into different mathematical contexts, demonstrating their versatility and power.