Problem 107 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\csc \theta=100$$

Step-by-Step Solution

Answer

Possible for some angle \(\theta\).

Step 1: Understanding the Problem

We need to determine whether the statement \(\csc \theta = 100\) is possible for some angle \(\theta\). The function \(\csc\theta\) is the cosecant of an angle, defined as the reciprocal of the sine function: \(\csc\theta = \frac{1}{\sin\theta}\).

Step 2: Considering the Range of Sine

The sine of an angle can only take values between -1 and 1, inclusive. This means \(\sin\theta\) must satisfy \(-1 \leq \sin\theta \leq 1\).

Step 3: Implication for Cosecant

Since \(\csc\theta = \frac{1}{\sin\theta}\), the absolute value of \(\csc\theta\) must be greater than or equal to 1 (\(|\csc\theta| \geq 1\)). This happens because when \(\sin\theta\) is close to 0, \(\csc\theta\) becomes very large.

Step 4: Checking the Given Condition

For \(\csc\theta = 100\), \(\sin\theta\) would have to be \(\frac{1}{100}\). Since \(\frac{1}{100}\) is within the range \(-1\) to \(1\), it's possible for some \(\theta\).

Step 5: Conclusion

Based on the value \(\csc\theta = 100\), and since \(\frac{1}{100}\) lies within the range of sine, it is possible for some angle \(\theta\). Therefore, the statement is possible.

Key Concepts

Cosecant function

The cosecant function is an important part of trigonometry, used to relate the angles inside a triangle to the lengths of the sides. It is the reciprocal of the sine function and is often abbreviated as "csc." This means that for an angle \( \theta \), the cosecant is defined as: \[ \csc\theta = \frac{1}{\sin\theta} \]This relationship explains why the cosecant function does not have values that are very close to zero.

**Understanding Cosecant Values**
The cosecant function can have values:

• Greater than or equal to 1
• Less than or equal to -1

The absolute value of \( \csc\theta \) always needs to be greater than or equal to 1 because \( \sin\theta \) can never be 0. For example, when \( \csc\theta = 100 \), we need \( \sin\theta \) to be quite small, such as \( \frac{1}{100} \).

The cosecant function is undefined when sine equals zero, which occurs at angles like \( 0, \pi, 2\pi, \ldots \). In these cases, \( \csc\theta \) would approach infinity.

Sine function

The sine function is one of the foundational trigonometric functions that we encounter in mathematics. It connects an angle in a right triangle to the ratio of two side lengths: the opposite side to the hypotenuse. The sine of an angle \( \theta \) is written as \( \sin\theta \).

**Range and Properties of Sine**
One key property of the sine function is its range. The values of \( \sin\theta \) are confined to the interval \([ -1, 1 ]\). This property is essential because it limits what values the sine function can take for any angle \( \theta \). For example, \( \sin\theta \) must be a fraction or decimal when \( \csc\theta \) is large, like in our case where \( \csc\theta = 100 \) implies \( \sin\theta = \frac{1}{100} \).

• Sine is continuous and follows a periodic pattern, repeating every \(2\pi\) radians.
• At particular angles like \( \theta = 90^\circ \) (or \( \pi/2\) radians), \( \sin\theta = 1 \).
• Similarly, at \( \theta = 270^\circ \) (or \( 3\pi/2\) radians), \( \sin\theta = -1 \).

The periodic nature gives us the ability to predict sine values across different angles simply by knowing a few key values.

Angle properties

Angles are a fundamental concept in geometry and trigonometry, providing a way to describe direction and rotation. In trigonometry, angles are often used to explore the functions like sine, cosine, and cosecant.

**Measuring Angles**
Angles can be expressed in degrees or radians, two different units of measurement. Often, radians are used in trigonometry because they capture the natural relationships between the circumference of a circle and its radius.

• 1 radian is approximately \( 57.2958 \) degrees.
• A full circle is \( 360^\circ \) or \( 2\pi \) radians.

For any given angle \( \theta \), the sine and cosecant values undergo periodic changes as \( \theta \) moves around the circle. This is why we see similar patterns in wave-like graphs produced by these functions. Whether in degrees or radians, understanding angles helps us use trigonometric functions to solve problems related to distance, height, and more.
Exploring these angle properties can guide us in determining whether certain trigonometric conditions, like \( \csc\theta = 100 \), are possible or impossible under normal circumstances.