Problem 6 - Precalculus with Limits

Question

Fill in the blank. \( \tan \left(u - v\right) \) =________

Step-by-Step Solution

Answer

The blank should be filled with \( \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)} \)

Step 1: Recall Trigonometric Identity

The formula for the tangent of the difference of two angles u and v is given as : \( \tan \left(u - v\right) \) = \(\frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}\)

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The blank should be filled with \( \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)} \)

Key Concepts

Trigonometric Identity

A trigonometric identity is an equation that is true for all values of the variables it contains. When you encounter a trigonometric problem involving the difference of angles, such as the exercise asking for the value of \( \tan(u - v) \), you are dealing with one of these powerful identities.

The specific identity used in this case is the tangent difference formula, which expresses \( \tan(u - v) \) as a quotient of the difference of the tangents of \( u \) and \( v \) and the sum of their products. That is:\[ \tan(u - v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)} \]

Understanding these identities is essential for simplifying and solving trigonometry problems, as they allow for the transformation of complex trigonometric expressions into simpler forms. Identities also help in verifying equations and proving various mathematical theorems.

Tan Function

The tan function, short for tangent, is one of the six fundamental trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, or as the sine function divided by the cosine function:

\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

The importance of the tan function in trigonometry cannot be overstated. It's particularly useful in situations involving angles of inclination, slopes, and whenever perpendicular and base aspects of a scenario are involved. The tan function is unique in that it's periodic with a period of \( \pi \), and has asymptotes, points where the function tends towards infinity, which occur at \( \frac{\pi}{2} \) and multiples thereof.

Trigonometry Problem Solving

Problem-solving in trigonometry often involves first identifying the relevant formulas and identities, as well as understanding the relationships between the trigonometric functions. It typically includes several steps, starting with outlining what is given and what needs to be found.

For example, in our given exercise, you are tasked with finding the expression for \( \tan(u - v) \). The first step is recognizing the appropriate identity to use, applying it, and then simplifying if necessary. It’s essential to also understand the domain and range of the trigonometric functions involved, as they have certain restrictions depending on the quadrant the angles are in.

Mastering trigonometry problem solving opens up pathways to advanced mathematics, including calculus and further analytical geometry, and also has applications in physics, engineering, and even computer science.