Problem 19
Question
Reciprocal Relationships. Evaluate to four decimal places. Find \(\csc \theta\) if \(\sin \theta=0.7352\)
Step-by-Step Solution
Verified Answer
\(\csc \theta \approx 1.3602\)
1Step 1: Understanding Cosecant Function
Recognize that the cosecant function is the reciprocal of the sine function. That is, \(\csc \theta = \frac{1}{\sin \theta}\).
2Step 2: Reciprocal of Sine Value
Given \(\sin \theta = 0.7352\), we can find \(\csc \theta\) by calculating the reciprocal of \(\sin \theta\), which is \(\frac{1}{0.7352}\).
3Step 3: Calculating Cosecant to Four Decimal Places
Use a calculator to find the reciprocal of \(0.7352\): \(\csc \theta = \frac{1}{0.7352} \approx 1.3602\) to four decimal places.
Key Concepts
Reciprocal RelationshipsSine FunctionTrigonometric Identities
Reciprocal Relationships
In trigonometry, one of the most fundamental concepts is that of reciprocal relationships between functions. These relationships are crucial as they enable us to derive one function from another. Take for instance the cosecant function, denoted as \( \csc \). This function is the reciprocal of the sine function (\( \sin \)), which means that \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
Understanding these relationships is not only important for solving trigonometric problems but also for simplifying expressions and proving trigonometric identities. It literally means that if we have the value of the sine function, inverting that value gives us the cosecant of the same angle. This simplicity is what makes studying reciprocal relationships an essential part of mastering trigonometry.
Understanding these relationships is not only important for solving trigonometric problems but also for simplifying expressions and proving trigonometric identities. It literally means that if we have the value of the sine function, inverting that value gives us the cosecant of the same angle. This simplicity is what makes studying reciprocal relationships an essential part of mastering trigonometry.
Sine Function
The sine function is a basic trigonometric function that defines a relationship between the length of the side opposite to an angle in a right triangle and the triangle's hypotenuse. It is written as \( \sin(\theta) \), where \( \theta \) is the angle in question.
In our case, we find that \( \sin \theta = 0.7352 \). This number is a ratio, which does not depend on the size of the specific right triangle but is constant for a given angle across all right triangles. This function oscillates between -1 and 1 and is foundational because it helps in defining other trigonometric functions, including the cosecant as discussed. Learning how to appropriately use the sine function is paramount for any student venturing into the realm of trigonometry.
In our case, we find that \( \sin \theta = 0.7352 \). This number is a ratio, which does not depend on the size of the specific right triangle but is constant for a given angle across all right triangles. This function oscillates between -1 and 1 and is foundational because it helps in defining other trigonometric functions, including the cosecant as discussed. Learning how to appropriately use the sine function is paramount for any student venturing into the realm of trigonometry.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. Identities like \( \csc(\theta) = \frac{1}{\sin(\theta)} \), discussed earlier, are fundamental tools in solving trigonometric equations and manipulating expressions.
It's important for students to not only memorize these identities but to understand how they are derived and connected to the geometric interpretations of trigonometry. Some common trigonometric identities include Pythagorean identities, angle sum and difference identities, and double angle formulas. Grasping these concepts equips learners with a powerful mathematical toolkit for a wide variety of applications including physics, engineering, and other stem fields.
It's important for students to not only memorize these identities but to understand how they are derived and connected to the geometric interpretations of trigonometry. Some common trigonometric identities include Pythagorean identities, angle sum and difference identities, and double angle formulas. Grasping these concepts equips learners with a powerful mathematical toolkit for a wide variety of applications including physics, engineering, and other stem fields.
Other exercises in this chapter
Problem 18
Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$238^{\circ}$$
View solution Problem 19
State whether the following expressions are positive or negative. Do not use your calculator, and try not to refer to your book. $$\sec 332^{\circ}$$
View solution Problem 21
Reciprocal Relationships. Evaluate to four decimal places. Find \(\sec \theta\) if \(\cos \theta=0.7354\)
View solution Problem 22
Give the algebraic signs of the sine, cosine, and tangent of the following. Do not use your calculator. $$110^{\circ}$$
View solution