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Chapter 1: Problem 53

True or false? Do not use a calculator. $$ \sin (23 \pi / 24)=-\sin (\pi / 24) $$

Short Answer

Expert verified
False

Step by step solution

01

Understand the properties of sine function

Recall the property of the sine function \(\sin(x) = \sin(\pi - x) = \sin(2\pi - x)\). This property helps in simplifying trigonometric expressions.
02

Express the angle in terms of common angles

Convert the given angle \(23\pi / 24\) into a form that uses properties of sine. Notice that \(23\pi / 24 = \pi - (\pi / 24)\).
03

Apply the sine property

Using the property \(\sin(x) = \sin(\pi - x)\), we get \(\sin(23\pi / 24) = \sin(\pi - \pi / 24) = \sin(\pi / 24)\).
04

Compare with the given expression

The problem states that \(\sin(23\pi / 24) = -\sin(\pi / 24)\). Since \(\sin(23\pi / 24) = \sin(\pi / 24)\), the equation \(\sin(23\pi / 24) = -\sin(\pi / 24)\) is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric identities
Trigonometric identities are fundamental tools used in trigonometry to manipulate and simplify expressions involving trigonometric functions. One key identity is the sine function property: \( \sin(x) = \sin(\pi - x) = \sin(2\pi - x) \).
This identity is useful because it helps you find equivalent angles and simplifies the process of solving trigonometric equations.

By using these identities, you can transform complex angles into more manageable forms. Whether you are dealing with sine, cosine, or tangent, understanding and applying these identities helps you resolve trigonometric problems efficiently.
angle conversion
Angle conversion is the process of changing the angle measurement from one form to another.
There are different units of measuring angles like degrees and radians. In trigonometry, radians are typically used.

For example, converting the angle \(23 \pi / 24 \) into a more familiar form involves recognizing that this angle can be expressed as \( \pi - (\pi / 24) \).
This step is critical because it sets the stage for applying your trigonometric identities effectively.

Knowing how to convert angles correctly lets you simplify calculations and compare angles more easily.
trigonometric functions
Trigonometric functions, such as sine, cosine, and tangent, describe relationships between the angles and sides of right-angled triangles.
The functions have specific properties and identities that are essential for solving equations.

In this exercise, you encounter the sine function \( \sin \). One of its properties states that \( \sin(x) = \sin( \pi - x) \).
This property was used to show that \( \sin(23\pi / 24)\ = \sin( \pi / 24) \).

Understanding the behavior of these functions and how they relate to angles is crucial in mastering trigonometry.

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Most popular questions from this chapter

Use a calculator to evaluate each expression. Round approximate answers to four decimal places. $$ \frac{\sin (\pi / 12)}{\cos (\pi / 12)} $$

Cooperative Learming In small groups, discuss the difference between radian measure and degree measure of an angle. Exactly how big is a unit circle? Why is a radian a real number? Could we define radian measure using any size circle?

Ordered Triples How many ordered triples of real numbers satisfy the following conditions? The product of \(x\) and \(y\) is \(z\). The product of \(x\) and \(z\) is \(y\). The product of \(y\) and \(z\) is \(x\).

True or false? Do not use a calculator. $$ \sin \left(210^{\circ}\right)=\sin \left(30^{\circ}\right) $$

Use a calculator to find the value of each function. Round answers to four decimal places. $$ \csc \left(49^{\circ} 13^{\prime}\right) $$
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