Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc 2 \theta=\frac{\csc \theta}{2 \cos \theta}$$

Step-by-Step Solution

Answer

The identity \(\csc 2 \theta=\frac{\csc \theta}{2 \cos \theta}\) is verified to be true both algebraically (by simplifying the expressions using trigonometric identities and equations) and through graphical representation.

Step 1: Recall Trigonometric Identity

We first recall the double angle formula for cosecant, which is \(\csc 2\theta = 2\csc \theta \cot \theta\). This will be useful in simplifying left side of the identity.

Step 2: Simplify Left Side

Substitute the double angle formula into the left side, we get \(\csc 2\theta = 2\csc \theta \cot\theta\). Since \(\cot \theta = \frac{1}{\tan \theta}\), and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), then \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), the equation becomes \(\csc 2\theta = 2\csc \theta \cdot \frac{\cos \theta}{\sin \theta}\).

Step 3: Simplify More

Since \(\csc \theta = \frac{1}{\sin \theta}\), replace it, we obtain \(\csc 2\theta = 2 \cdot \frac{1}{\sin \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{1}{2\sin^2\theta}\cdot2\cos \theta = \frac{\csc \theta}{2\cos \theta}\). Left side and right side are equal now, the trigonometric identity is proved.

Step 4: Verify by Graphing

Proceed to verify graphically using a graphing utility. Graph both \(\csc 2\theta\) and \(\frac{\csc \theta}{2\cos \theta}\) on the same graph. They should produce identical graphs if the identity holds true.

Key Concepts

Cosecant Function

The cosecant function, indicated as \( \csc \theta \), is one of the trigonometric functions related to a right-angled triangle. It is defined as the reciprocal of the sine function. Therefore, \( \csc \theta = \frac{1}{\sin \theta} \). This function becomes particularly useful in various trigonometric identities and transformations.

• **Sin and Cosecant:** Since cosecant is the reciprocal of sine, it is undefined wherever sine is zero. This occurs at integer multiples of \( \pi \).
• **Range and Domain:** Because \( \csc \theta \) is the reciprocal, its range is \((-\infty, -1] \cup [1, \infty)\). Its domain consists of all real numbers except where \( \sin \theta = 0 \).
• **Important Property:** For any angle \( \theta \), \( \csc \theta \) changes signs whenever sine changes signs.

Understanding the cosecant function provides a foundation for solving trigonometric identities, like the one in our problem.

Double Angle Formulas

Double angle formulas are essential tools in trigonometry that help transform expressions or solve trigonometric equations. The double angle formula for cosecant, \( \csc 2\theta \), can be derived using known relationships among trigonometric functions.

• **The Formula:** Specifically, \( \csc 2\theta = 2\csc \theta \cot \theta \). To remember this, consider translating combined angles in terms of single angle functions.
• **Breaking it Down:** The cotangent function can be expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, substituting \( \cot \theta \) gives \( \csc 2\theta = 2 \times \frac{1}{\sin \theta} \times \frac{\cos \theta}{\sin \theta} \).
• **Application:** These formulas simplify complex expressions, like those encountered in calculus and physics, providing simpler representations that are easier to work with analytically or graphically.

Having a handle on these formulas, like used in this exercise, can significantly ease the process of proving or simplifying trigonometric identities.

Graphing Trigonometric Functions

Graphing trigonometric functions can visualize the validity of expressions and identities. For verifying \( \csc 2\theta = \frac{\csc \theta}{2\cos \theta} \), graphing the two sides of this identity should produce identical curves if the identity is true.

• **Graph Setup:** Use a graphing calculator or utility, input both functions, \( \csc 2\theta \) and \( \frac{\csc \theta}{2\cos \theta} \), possibly as \( y_1 \) and \( y_2 \).
• **Observation:** If both functions superimpose exactly over each other over the domain defined, then it visually confirms the algebraic identity.
• **Aspects of Graphs:** Notice the reflection and symmetry in cosecant graphs. They generally have vertical asymptotes corresponding to the zeros of the sine function, giving them a distinct U-like shape.

Graphing extends beyond mere visuals; it serves as crucial confirmation of trigonometric truths explored numerically or symbolically.