Problem 71 - Precalculus with Limits

Question

In Exercises 71 - 74, simplify the expression algebraically and use a graphing utility to confirm your answer graphically. \( \cos\left(\dfrac{3\pi}{2} - x\right) \)

Step-by-Step Solution

Answer

The simplified form of \( \cos\left(\dfrac{3\pi}{2} - x\right) \) is \( - \sin(x) \).

Step 1: Apply cosine difference identity

Substitute the given function into the difference identity formula for cosine: \( \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b) \), not forgetting that a is \( \dfrac{3\pi}{2} \) and b is \( x \). This gives the expanded form \( \cos\left(\dfrac{3\pi}{2}\right) \cdot \cos(x) + \sin\left(\dfrac{3\pi}{2}\right) \cdot \sin(x) \).

Step 2: Substitute values for cosine and sine

Since \( \cos\left(\dfrac{3\pi}{2}\right) = 0 \) and \( \sin\left(\dfrac{3\pi}{2}\right) = -1 \), substitute these values into the expanded equation we have from step 1 which gives the result as: \(0 \cdot \cos(x) - 1 \cdot \sin(x) \)

Step 3: Simplify expression

On simplifying the expression, \(0 \cdot \cos(x) - 1 \cdot \sin(x) \) becomes \( - \sin(x) \). we obtain the simplified algebraic form.

Step 4: Graphical representation

To confirm the algebraic solution, plot both the original function \( \cos\left(\dfrac{3\pi}{2} - x\right) \) and the simplified expression \( - \sin(x) \) on a graph. If both representations match, it confirms that the algebraic simplification is correct.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities allow us to simplify complex expressions and solve trigonometric equations more easily. One such identity is the cosine difference identity, \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \), which expresses the cosine of the difference between two angles in terms of their sine and cosine values. In the exercise, when the angle \( \frac{3\pi}{2} \) is involved, we know that \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \), which simplifies the expression considerably. This identity is pivotal in algebraically manipulating expressions to their simplest form.

It is essential for students to memorize the basic trigonometric identities, as they will serve as tools for tackling a variety of problems in trigonometry, calculus, and even complex number theory.

Graphing Utilities

Graphing utilities, such as graphing calculators or software programs, are incredibly useful for confirming the results of trigonometric functions and identities. Upon simplifying an expression algebraically, you can visualize the result graphically to ensure that the simplification is correct. In our exercise, the original function \( \cos\left(\frac{3\pi}{2} - x\right) \) and the resulting simplified function \( -\sin(x) \) can both be plotted using these tools. When both graphs coincide, it serves as a visual confirmation of the algebraic work.

Students are encouraged to become familiar with various graphing utilities, as they can deepen their understanding of how trigonometric functions behave and interact. This hands-on approach can solidify comprehension and make abstract concepts more tangible.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics, effectively reducing them to the most basic form without changing their value. Simplification often involves combining like terms, factoring, expanding, and employing trigonometric identities. In the given exercise, the expression \( \cos\left(\frac{3\pi}{2} - x\right) \) is simplified by applying the cosine difference identity and substituting the known values of sine and cosine for \( \frac{3\pi}{2} \). This reduces a complex trigonometric function to a simple one, \( -\sin(x) \).

When dealing with trigonometric expressions, always look for opportunities to use known identities to condense the expression. This can make problems less daunting and more manageable, especially when tackling more complicated equations or when integrating and differentiating in calculus.

Confirming Solutions Graphically

Confirming solutions graphically is akin to adding a second layer of verification to your algebraic solutions. By plotting the original problem and the algebraically simplified expression on the same graph, you can visually inspect if the two overlap or coincide. This graphical confirmation offers immediate feedback and is particularly helpful when algebraic manipulation yields an unexpected form. As in our textbook exercise, after simplifying \( \cos\left(\frac{3\pi}{2} - x\right) \) to \( -\sin(x) \), graphing both functions should yield the same line, thereby confirming the simplification's validity.

Educators often advise students to use graphing utilities as a means to double-check their work. It is a valuable technique in not only ensuring accuracy but also in developing a student's intuition for the behavior of mathematical functions.