Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\csc \theta, \text { given that } \sin \theta=-\frac{3}{7}$$

Step-by-Step Solution

Answer

\( \csc \theta = -\frac{7}{3} \)

Step 1: Identify the Reciprocal Identity

The cosecant function, \( \csc \theta \), is the reciprocal of the sine function. The reciprocal relationship is defined as \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute the Known Value

Given \( \sin \theta = -\frac{3}{7} \), substitute \( \sin \theta \) into the reciprocal identity: \( \csc \theta = \frac{1}{-\frac{3}{7}} \).

Step 3: Simplify the Expression

To find the reciprocal, flip the fraction. Thus, \( \csc \theta = -\frac{7}{3} \). There is no need to rationalize the denominator, as there is no square root involved.

Key Concepts

Reciprocal Identities

Reciprocal identities are an essential concept in trigonometry. They relate trigonometric functions to each other by defining one function as the inverse of another. Specifically, for basic trigonometric functions like sine, cosine, and tangent, each has a reciprocal function. These identities can be incredibly useful for simplifying expressions and solving trigonometric equations.

The main reciprocal identities include:

• The cosecant function: \( \csc \theta = \frac{1}{\sin \theta} \)
• The secant function: \( \sec \theta = \frac{1}{\cos \theta} \)
• The cotangent function: \( \cot \theta = \frac{1}{\tan \theta} \)

Understanding these identities helps you manipulate and solve trigonometric equations more easily. It's like having a key to unlock the relationships between trig functions. Using reciprocal identities, you can convert complex expressions into simpler forms.

Cosecant Function

The cosecant function, often denoted as \( \csc \theta \), is one of the six fundamental trigonometric functions and is the reciprocal of the sine function. Knowing this relationship allows you to find \( \csc \theta \) if you know \( \sin \theta \).

To compute \( \csc \theta \), you use the formula:\[\csc \theta = \frac{1}{\sin \theta}\]This means if you are given that \( \sin \theta = -\frac{3}{7} \), you can find \( \csc \theta \) by simply flipping the fraction:\[\csc \theta = \frac{1}{-\frac{3}{7}} = -\frac{7}{3}\]Such a result highlights how reciprocal functions operate and reflect the idea of inversion.

• If \( \sin \theta \) is positive, \( \csc \theta \) will also be positive.
• If \( \sin \theta \) is negative, as in this example, \( \csc \theta \) will be negative as well.

The cosecant function is particularly useful in problems involving triangles, waves, and oscillations.

Sine Function

The sine function, one of the most familiar trigonometric functions, is symbolized by \( \sin \theta \). It relates to the ratio of the length of the opposite side to the hypotenuse in a right triangle.pose Additionally, it serves as the foundation for periodic functions such as waves, describing cycles in time and space.

Graphically, \( \sin \theta \) represents a smooth, wave-like curve that oscillates between 1 and -1 as \( \theta \) varies. This periodicity is significant in fields such as physics and engineering, notably in phenomena like sound waves and alternating currents.

• \( \sin \theta = 1 \) at \( \theta = \frac{\pi}{2} \)
• \( \sin \theta = 0 \) at \( \theta = 0, \pi, 2\pi \), etc.
• \( \sin \theta = -1 \) at \( \theta = \frac{3\pi}{2} \)

In the context of reciprocal identities, knowing \( \sin \theta \) allows you to find the cosecant using its reciprocal form, enhancing your ability to solve various trigonometric problems. Whether applied in angles, triangles, or waves, understanding \( \sin \theta \) is crucial for efficiently tackling exercises involving trigonometric identities.