Problem 28 - Contemporary Precalculus

Question

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\cot \theta=-2.4$$

Step-by-Step Solution

Answer

a. 112.5° b. 202.5° c. 247.5° d. 67.5°

Step 1: The cotangent function is defined as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). To find the principal angle, we want to find the angle whose cotangent is equal to -2.4. We can use the arccotangent function for this:$$\theta = \operatorname{arccot}(-2.4)$$Remember to use your calculator in degree mode, and replace \(\pi\) with \(180^{\circ}\) if necessary. #Step 2: Find the principal angle within the specified range#

Once you've found the arccot(-2.4), make sure your angle is within the specified range of \(0^{\circ} \leq \theta < 360^{\circ}\). If it is not, add or subtract multiples of \(360^{\circ}\) until it lies within the desired range. #Step 3: Use the periodicity of the cotangent function to find additional solutions#

Step 2: The cotangent function is periodic with a period of \(180^{\circ}\). So, if \(\theta_1\) is a solution, then \(\theta_2 = \theta_1 \pm 180^{\circ}\) will also be a solution. Keep adding or subtracting \(180^{\circ}\) until you find all the angles within the specified range. #Step 4: List all the angles within the specified range that fulfill the equation#

After finding all the angles within the specified range that satisfy the equation, list them as your final answer. And don't forget to give your answer in degrees, since the problem specifies degrees.

Key Concepts

Cotangent Function

The cotangent function, often denoted as \( \cot \theta \), is a fundamental concept in trigonometry. It is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, or more precisely, it is the reciprocal of the tangent function. Mathematically, this is represented as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This function is very useful in solving various trigonometric equations. In these problems, sometimes we need to find the angle \( \theta \), which when substituted in the expression gives a specific cotangent value.
To find such an angle manually over a given range, you might need to use the arccotangent (inverse cotangent) function on a calculator set to degree mode. This ensures the result is given as an angle measured in degrees, which is crucial for problems where the angle is specified in degrees.
Understanding the functionality and definition of the cotangent helps in deciphering how to manipulate and solve equations that involve this trigonometric function.

Periodicity in Trigonometry

Periodicity is a core attribute of trigonometric functions, including the cotangent function. It refers to the repeating nature of these functions at regular intervals, called periods. For the cotangent function, this period is \(180^\circ\). This means that if \( \theta \) is a solution to a cotangent equation, then \( \theta \pm 180^\circ \) will also be a solution.
This property is essential when solving equations like \( \cot \theta = -2.4 \) and finding all possible angles \( \theta \) within the range of \( 0^\circ \) to \( 360^\circ \). Once you have determined a principal angle using arccotangent, adding or subtracting multiples of \( 180^\circ \) will give additional solutions.

• This systematic approach ensures all possible solutions within a specified interval are identified.
• Recognizing periodicity can simplify solving and verifying trigonometric identities and equations.

Understanding periodicity can significantly aid in mastering trigonometric problems as it allows you to predict and confirm additional solutions efficiently.

Angle Measurement in Degrees

Angle measurement can be done using various units, but degrees are one of the most commonly used, especially in educational contexts. In the degree system, a full circle is divided into \(360^\circ\). This unit is very intuitive as it reflects the natural way we view objects and angles in daily life.

• Degrees provide a straightforward and easy-to-understand way of expressing angle size.
• When solving problems, it is key to ensure calculations are performed in degree mode, especially when using calculators that can work in radians or degrees.

For trigonometric problems involving equations like \( \cot \theta = -2.4 \), angles need to be consistently expressed and calculated in this unit.
This is crucial for ensuring accurate results within the specified range. Understanding degrees and being able to convert between different angle measures is foundational for mastering trigonometry and its applications, and it helps maintain precision and clarity in problem-solving.