Problem 7 - Algebra 2

Question

Verify each identity. $$ \sin \theta \csc \theta=1 $$

Step-by-Step Solution

Answer

After substitifying \(\csc \theta\) by its equivalent \(1/\sin \theta\) and simplifying \(\sin \theta * 1/\sin \theta\), the result is indeed 1 as required, thus verifying the given identity.

Step 1: Recall the definition

First, recall the definition of the function \(\csc \theta\) in terms of the function \(\sin \theta\). By definition, \(\csc \theta = 1/\sin \theta\).

Step 2: Substitute the expression

Substitute \(\csc \theta\) with its equivalent, \(1/\sin \theta\), in the expression to be verified. So, \(\sin \theta \csc \theta = \sin \theta * 1/\sin \theta \)

Step 3: Simplify the expression

Simplify the right-hand side of the equation. Cancelling the common factor of \(\sin \theta\) at numerator and denominator gives the result 1.

Key Concepts

Sine Function

The sine function, commonly denoted as \( \sin \theta \), is a fundamental aspect of trigonometry. It identifies the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The function is pivotal in describing wave-like phenomena, such as sound and light waves, using the unit circle approach.
In the unit circle, the sine of an angle \( \theta \) is the \( y \)-coordinate of the corresponding point on the circle's circumference. Typically, its values range from -1 to 1 as the angle traverses through the circle from 0 to 360 degrees or 0 to \( 2\pi \) radians. Key sine values include:

• \( \sin 0 = 0 \)
• \( \sin \frac{\pi}{2} = 1 \)
• \( \sin \pi = 0 \)
• \( \sin \frac{3\pi}{2} = -1 \)
• \( \sin 2\pi = 0 \)

Understanding the sine function is instrumental for verifying trigonometric identities and solving trigonometric equations. It provides the foundational bridge needed to explore relationships between different trigonometric entities.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function. In simpler terms, it is defined as \( \csc \theta = \frac{1}{\sin \theta} \). This relationship means whenever the sine of an angle is known, the cosecant can be easily determined by taking its reciprocal.
It's crucial to note that \( \csc \theta \) is undefined when \( \sin \theta = 0 \) because division by zero is not possible. Therefore, the cosecant function has vertical asymptotes at these points, leading to its discontinuity at specific angles, such as 0, \( \pi \), and \( 2\pi \).
When graphically represented, the cosecant function demonstrates distinct properties due to its reciprocal nature; steep rises and falls occur where the sine function approaches zero. For instance, when \( \sin \theta \) reaches its minimum or maximum, \( \csc \theta \), inversely, tends to infinity. These complementary characteristics are essential in the broader landscape of trigonometric identities.

Simplification

Simplification is a crucial technique in mathematics that involves rewriting expressions in their simplest or most concise form. The process often requires identifying and canceling out common factors within numerators and denominators, thereby transforming complex expressions into more manageable forms.
In the given exercise, the identity \( \sin \theta \csc \theta = 1 \) illustrates an elegant simplification. By substituting \( \csc \theta \) with its equivalent expression \( \frac{1}{\sin \theta} \), the equation becomes \( \sin \theta \times \frac{1}{\sin \theta} \). Here, \( \sin \theta \) and its reciprocal naturally cancel each other out, simplifying the expression to \( 1 \).
Employing such simplifications helps not only in verifying trigonometric identities but also in solving more complex algebraic and trigonometric problems. By breaking down expressions, simplification enhances both comprehension and computational efficiency. It's a valuable tool for students in navigating and mastering mathematical challenges.