Problem 6 - Precalculus

Question

Carry out the indicated operations. (a) \((3-2 T)^{2}\) (b) \((3-2 \tan \theta)^{2}\)

Step-by-Step Solution

Answer

(a) \(9 - 12T + 4T^2\); (b) \(9 - 12\tan\theta + 4\tan^2\theta\).

Step 1: Unpack the Square

The expression \((3 - 2T)^2\) can be expanded. Recall that \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 3\) and \(b = 2T\). The expression becomes \(3^2 - 2(3)(2T) + (2T)^2\).

Step 2: Calculate Each Term

Compute each term separately: \(3^2 = 9\), \(-2(3)(2T) = -12T\), and \((2T)^2 = 4T^2\).

Step 3: Combine the Terms

Combine the results from Step 2: the expression becomes \(9 - 12T + 4T^2\).

Step 4: Unpack the Square with Trigonometric Function

Similarly, for \((3 - 2\tan\theta)^2\), use the same identity \((a - b)^2 = a^2 - 2ab + b^2\). Here \(a = 3\) and \(b = 2\tan\theta\), resulting in \(3^2 - 2(3)(2\tan\theta) + (2\tan\theta)^2\).

Step 5: Calculate Each Term with Trigonometry

Compute each term: \(3^2 = 9\), \(-2(3)(2\tan\theta) = -12\tan\theta\), and \((2\tan\theta)^2 = 4\tan^2\theta\).

Step 6: Combine Terms with Trigonometry

The expanded expression becomes \(9 - 12\tan\theta + 4\tan^2\theta\).

Key Concepts

Polynomial Operations

Polynomial operations involve performing calculations on polynomial expressions, which include adding, subtracting, multiplying, and even dividing polynomials. The exercise above is a classic example that focuses on expanding a binomial expression. A binomial is a polynomial with two terms, like

• \((3-2T)^2\)
• \((3-2 \tan \theta)^2\)

To expand these, we apply the formula for squaring a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). This formula helps in breaking down the square into simpler arithmetic operations.
For example, in part (a), we identify \(a = 3\) and \(b = 2T\). Substituting these into the formula, we get:

• \[ 3^2 - 2(3)(2T) + (2T)^2 = 9 - 12T + 4T^2 \]

The computation of each individual term, such as \((2T)^2 = 4T^2\), is a typical polynomial operation where each element of the binomial is squared and the cross-term is multiplied by 2 and then simplified.
Understanding these operations builds the foundation for more complex polynomial manipulations seen later in algebraic studies.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They play a crucial role when dealing with algebraic expressions involving trigonometry.
In the exercise, when expanding the expression \((3 - 2 \tan \theta)^2\), we make use of trigonometric identities indirectly. The primary identity in use is still the binomial \((a - b)^2\) formula, which provides the structure for expansion.

• \(a = 3\)
• \(b = 2 \tan \theta\)

By applying this expansion formula, we get

• \[3^2 - 2(3)(2 \tan \theta) + (2 \tan \theta)^2 = 9 - 12 \tan \theta + 4 \tan^2 \theta\]

This process illustrates how algebraic techniques can work hand in hand with trigonometric concepts.
The term \(4 \tan^2 \theta\) is derived by squaring \(2 \tan \theta\), showcasing how easily trigonometric functions integrate into algebraic expressions. Grasping these expansions can greatly aid in simplifying complex trigonometric expressions later in mathematics.

Algebraic Expressions

Algebraic expressions are mathematical statements that combine numbers, variables, and operations. These expressions are prevalent in algebra, serving as the building blocks of equations and functions.
In the problem we see above, the expression \((3 - 2T)^2\) is an algebraic expression that contains both a constant (3) and a variable term \(-2T\). When expanded, it results in the polynomial \(9 - 12T + 4T^2\). This expansion highlights how algebraic expressions evolve through operations like squaring.
Similarly, for \((3 - 2 \tan \theta)^2\), even though it involves a trigonometric function, it's still treated as an algebraic expression because it consists of variables and constants. The expansion leads to \(9 - 12 \tan \theta + 4 \tan^2 \theta\), further revealing the seamless integration of trigonometric terms in algebraic operations.
Key points to understanding algebraic expressions include:

• Recognizing and manipulating different components, such as coefficients and variables.
• Performing operations like addition, subtraction, and multiplication sensibly.
• Applying algebraic properties and formulas for simplification.

These skills are crucial in efficiently solving algebra problems and effectively writing expressions for complex real-world problems.