Problem 47 - Algebra and Trigonometry

Question

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\frac{\tan t \sin t}{\cot t}, \quad\) quadrant III

Step-by-Step Solution

Answer

The expression is negative.

Step 1: Understanding Quadrant III

In Quadrant III, both sine and tangent functions are negative. Cotangent, being the reciprocal of tangent, is also negative.

Step 2: Expressing Trigonometric Functions

Express the given expression in terms of sine and cosine functions: \( \frac{\tan t \cdot \sin t}{\cot t} = \frac{\left(\frac{\sin t}{\cos t}\right) \cdot \sin t}{\frac{\cos t}{\sin t}} \). This simplifies to \( \frac{\sin^2 t}{\cos t} \cdot \frac{\sin t}{\cos t} \).

Step 3: Simplifying the Expression

Now simplify: \( \frac{\sin^2 t \cdot \sin t}{\cos t \cdot \cos t} = \frac{\sin^3 t}{\cos^2 t} \).

Step 4: Determining the Sign of the Expression

Since both \(\sin t\) and \(\cos t\) are negative in Quadrant III, \(\sin^3 t\) (the cube of a negative number) is negative, and \(\cos^2 t\) (the square of a negative number) is positive. Thus, the overall expression \( \frac{\sin^3 t}{\cos^2 t} \) is negative.

Key Concepts

Quadrant Analysis

When dealing with trigonometric functions, knowing the quadrant of the terminal point is essential to determine the sign of the expression. Each quadrant has specific characteristics:

• In Quadrant III, both sine (\(\sin t\)) and tangent (\(\tan t\)) are negative.
• Coscine (\(\cos t\)) is also negative because cosine is negative in Quadrants II and III.
• Cotangent (\(\cot t\)), being the reciprocal of \(\tan t\), remains negative.

Understanding this allows us to predict the behavior of the trig functions without explicitly calculating the values. This understanding is crucial for determining whether the resulting trigonometric expression is positive or negative.

Trigonometric Identities

Trigonometric identities are essential tools for transforming and simplifying expressions. In this exercise, we used these identities:

• The tangent function: \(\tan t = \frac{\sin t}{\cos t}\).
• The cotangent function: \(\cot t = \frac{\cos t}{\sin t}\).

By substituting these into the given expression \(\frac{\tan t \sin t}{\cot t}\), conversions simplify the operations by expressing all terms in forms that can cancel or combine easily. Rewriting trigonometric expressions using identities aids in managing complex equations, making analysis more straightforward.

Expression Simplification

Simplifying expressions involves reducing them to their simplest form while preserving equivalency. Let’s see this in action:First, express the given trigonometric expression:

• We rewrite \(\frac{\tan t \cdot \sin t}{\cot t}\) as \(\frac{\left(\frac{\sin t}{\cos t}\right) \cdot \sin t}{\frac{\cos t}{\sin t}}\).
• Multiply to get \(\frac{\sin^2 t}{\cos t} \cdot \frac{\sin t}{\cos t} = \frac{\sin^3 t}{\cos^2 t}\).

Simplification not only removes complexities but provides clarity to the expression’s behavior. Knowing the signs, as explained earlier, the final form \(\frac{\sin^3 t}{\cos^2 t}\) evaluates to negative in Quadrant III. This type of simplifying helps in solving more advanced problems easily.