Problem 243 - Precalculus

Question

For the following exercises, use identities to evaluate the expression. If \(\cot (t) \approx 0.58, \quad\) and \(\cos (t) \approx 0.5, \quad\) find \(\csc (t)\)

Step-by-Step Solution

Answer

\( \csc(t) \approx 1.16 \)

Step 1: Understand the Given Information

You are given \( \cot(t) \approx 0.58 \) and \( \cos(t) \approx 0.5 \). We need to find \( \csc(t) \). Use trigonometric identities to connect these values.

Step 2: Use the Reciprocal Identity for Cotangent

Recall the identity relating cotangent to tangent: \( \cot(t) = \frac{1}{\tan(t)} \). We can find \( \tan(t) \) using \( \tan(t) = \frac{1}{\cot(t)} \). Substitute the given \( \cot(t) \approx 0.58 \): \[\tan(t) \approx \frac{1}{0.58} \approx 1.724\]

Step 3: Use the Trigonometric Identity for Tangent with Sine and Cosine

We know \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). Substitute \( \tan(t) \approx 1.724 \) and \( \cos(t) \approx 0.5 \) to find \( \sin(t) \):\[1.724 = \frac{\sin(t)}{0.5}\]Solving for \( \sin(t) \):\[\sin(t) \approx 1.724 \times 0.5 = 0.862\]

Step 4: Use the Reciprocal Identity for Cosecant

The cosecant is the reciprocal of the sine: \( \csc(t) = \frac{1}{\sin(t)} \). Substitute \( \sin(t) \approx 0.862 \) to find \( \csc(t) \):\[\csc(t) \approx \frac{1}{0.862} \approx 1.16\]

Step 5: Conclude the Calculation

Having found \( \csc(t) \approx 1.16 \), we used the given values and trigonometric identities to evaluate the expression sufficiently.

Key Concepts

cotangent

The cotangent is a fundamental trigonometric function related to angles within a triangle or on the unit circle. It is often denoted as \( \cot(t) \) and defined through the ratio of the adjacent side over the opposite side in a right triangle. Mathematically, it can be expressed as:

• \( \cot(t) = \frac{1}{\tan(t)} \)
• \( \cot(t) = \frac{\cos(t)}{\sin(t)} \)

This dual definition allows for flexibility when solving problems since you can switch between using the tangent, sine, or cosine functions.

In the exercise, knowing \( \cot(t) \approx 0.58 \) allows us to find \( \tan(t) \) using the reciprocal identity \( \tan(t) = \frac{1}{\cot(t)} \). Mastering these identities will significantly simplify operations and analyses involving trigonometric expressions.

cosecant

Cosecant is another key trigonometric function, symbolized as \( \csc(t) \). It is particularly crucial as it relates inversely to the sine of an angle. The definition stands as:

• \( \csc(t) = \frac{1}{\sin(t)} \)

Understanding and using the notion of reciprocals is central to trigonometry.

In problem-solving, once we have determined \( \sin(t) \), finding \( \csc(t) \) becomes straightforward. By using \( \sin(t) \approx 0.862 \), the exercise shows that \( \csc(t) \approx \frac{1}{0.862} \). Grasping this identity can clarify many trigonometric equations and ensures accurate calculations.

sine and cosine relationship

The relationship between sine and cosine is foundational in trigonometry, forming the basis for many identities and equations. These two functions are interrelated through the Pythagorean identity:

• \( \sin^2(t) + \cos^2(t) = 1 \)

This identity allows us to find one function if the other is known, provided the angle lies within a known range. In the unit circle framework, \( \sin(t) \) represents the y-coordinate, while \( \cos(t) \) stands for the x-coordinate of a point. Their interplay is critical in understanding the behavior of angles and in translating complex algebraic problems into geometric interpretations.

In the given exercise, knowing \( \cos(t) \approx 0.5 \) and \( \tan(t) \approx \frac{\sin(t)}{\cos(t)} \), we can resolve \( \sin(t) \approx 0.862 \). This understanding shows how tightly interconnected these trigonometric functions are.