Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\csc ^{2} \theta \sin ^{2} \theta=1$$

Step-by-Step Solution

Answer

The given equation simplifies to \(1 = 1\), thus validating its identity for the given range of \( \theta \.

Step 1: Identifying and Using Trigonometric Identities

The given equation is \(\csc ^{2} \theta \sin ^{2} \theta=1\). As stated in the analysis, the key identity to use here is \(\csc \theta = 1/\sin \theta \). Hence, \(\csc ^{2} \theta = 1/(\sin^2 \theta)\). Substituting this into the equation, we get: \((1/\sin^2 \theta) \cdot \sin^2 \theta = 1\).

Step 2: Simplifying the Equation

The equation \(1/\sin^2 \theta \cdot \sin^2 \theta = 1\) simplifies quickly. Since \(\sin^2 \theta\) in the denominator and numerator cancel each other, the equation simplifies to \(1 = 1\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. This relationship can be expressed as \( \csc \theta = \frac{1}{\sin \theta} \). Understanding this reciprocal identity is crucial when working with trigonometric equations.
In mathematical problems, the cosecant function is often used to simplify complex equations by transforming them into forms that are easier to manipulate and solve. For example, if you see \( \csc^2 \theta \), you can rewrite it as \( \frac{1}{\sin^2 \theta} \). This transformation often makes subsequent algebraic steps more straightforward.
To reinforce your understanding:

• Remember that the cosecant is undefined for angles where \( \sin \theta = 0 \).
• The cosecant function is periodic, similar to the sine function, and it repeats its values in regular intervals.

Sine Function

The sine function, represented as \( \sin \theta \), is a fundamental building block in trigonometry. It gives the y-coordinate of a point on the unit circle corresponding to an angle \( \theta \). Understanding its behavior is crucial when interpreting trigonometric identities or solving equations.
When studying this function, it's helpful to remember some of its key properties:

• The sine function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \) units.
• It ranges between -1 and 1; thus, \(-1 \leq \sin \theta \leq 1\).
• At \( \theta = 0 \) and \( \theta = \pi \), \( \sin \theta \) is 0; this directly affects the cosecant function since it's undefined at these points.

Knowing these properties aids in evaluating and solving identities like \( \csc^2 \theta \sin^2 \theta = 1 \), where sine's reciprocal role comes into play.

Simplifying Trigonometric Equations

Simplifying trigonometric equations is an essential skill in mathematics. It involves transforming complex expressions into simpler or more workable forms using identities and algebraic manipulation.
In the original exercise, the equation \( \csc^2 \theta \sin^2 \theta = 1 \) was given. To simplify this:

• Recognize that \( \csc^2 \theta \) can be replaced by \( \frac{1}{\sin^2 \theta} \) using the reciprocal identity.
• Once substituted into the equation, \( \frac{1}{\sin^2 \theta} \cdot \sin^2 \theta \) simplifies to \( 1 \) as the \( \sin^2 \theta \) terms cancel each other.

This illustrates how using known identities and properties leads to simplification, reducing complex expressions to their simplest forms. Practicing these techniques helps build confidence and proficiency in handling trigonometric equations.