Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\sec \theta \cot \theta \sin \theta$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Express TRIG Functions in Sine and Cosine

First, express each trigonometric function in the expression in terms of sine and cosine. We have:- \( \sec \theta = \frac{1}{\cos \theta} \)- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)Thus, the expression \( \sec \theta \cot \theta \sin \theta \) becomes \( \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \cdot \sin \theta \).

Step 2: Simplify the Expression

Now, simplify the expression:Substitute the expressions from Step 1:\[ \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \cdot \sin \theta \]Cancel \( \cos \theta \) with \( \cos \theta \) from the numerator and denominator:\[ \frac{1}{\sin \theta} \cdot \sin \theta \]Since \( \sin \theta \) cancels with the same term in the numerator and denominator:The expression simplifies to 1.

Key Concepts

Sine and Cosine

Sine and cosine are two of the most fundamental trigonometric functions used in mathematics. They help describe the relationship between the angles and sides of a right triangle.
- **Sine Function (\( \sin \theta \)):** This function gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is represented as \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \).
- **Cosine Function (\( \cos \theta \)):** This function represents the ratio of the adjacent side to the hypotenuse. Its formula is \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
These functions are not only useful for solving triangles but also play a vital role in wave functions, oscillations, and many real-life applications. For the original expression involving trigonometric identities, writing everything in terms of \( \sin \theta \) and \( \cos \theta \) makes it easier to simplify.

Simplification

Simplification in mathematics is all about reducing an expression to its simplest form. This involves eliminating any unnecessary components from the expression. In terms of trigonometric expressions, it is often helpful to express all trigonometric functions in terms of sine and cosine. This can often reveal opportunities to simplify the expression.
For example, in the exercise \( \sec \theta \cot \theta \sin \theta \):

• We converted \( \sec \theta \) to \( \frac{1}{\cos \theta} \)
• We converted \( \cot \theta \) to \( \frac{\cos \theta}{\sin \theta} \)

The expression then becomes a straightforward series of cancellations, eventually resulting in the expression simplifying to 1. The simplification process often involves canceling terms that appear both in the numerator and the denominator.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. Beyond sine and cosine discussed earlier, several other trigonometric functions are commonly used:
- **Secant (\( \sec \theta \)):** This is the reciprocal of cosine, given by \( \sec \theta = \frac{1}{\cos \theta} \).- **Cotangent (\( \cot \theta \)):** The reciprocal of tangent, represented as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
These functions are commonly used in various mathematical calculations, especially when dealing with right-angled triangles. Understanding these trigonometric identities and their relationships with one another can simplify complex expressions and solve real-world problems with ease.