Show that the following equations are not identities. √(sin^2 x-9)=sinx-3

Name: Problem 62, Chapter 6: Precalculus
Uploaded: 2020-05-31T15:32:10-08:00
Duration: 1 min 54 s
Channel: Andrew Bassila
Description: Show that the following equations are not identities. √(sin^2 x-9)=sinx-3

step 1 All right, we're tasked again here to show that this is not an identity. And I really like this

Show that the following equations are not identities....

Key Concepts

Domain Restrictions

This concept involves determining the set of all possible input values (in this case, values of x) for which the expressions in the equation are defined. When dealing with functions like square roots, it is necessary to ensure that the expression under the radical sign meets the criteria required for the function to output real numbers. For example, expressions inside a square root must be non-negative when working in the real number system.

Properties of Square Roots

Square roots in the real number system always yield non-negative results, which means that the output of a square root function cannot be negative. This property is critical when solving equations because it imposes restrictions on possible solutions and helps to determine if both sides of an equation can ever be equal under given circumstances.

Identities vs. Conditional Equations

An identity is an equation that holds true for all values within the domain of the variables involved, whereas a conditional equation is true only for specific values of the variables. Understanding the difference between these helps in analyzing statements to determine whether an equation is universally valid (an identity) or only valid under certain conditions.

Range of Trigonometric Functions

Trigonometric functions, such as the sine function in this context, have specific ranges. The sine function, for example, has outputs confined to the interval [-1, 1]. Recognizing the range is essential when these functions interact with other operations or functions, as it can lead to inherent restrictions in an equation and affect whether an identity can hold.

Extraneous Solutions

This concept refers to solutions that emerge from the process of solving an equation but do not satisfy the original equation. Especially in equations involving radical operations, the method of squaring both sides may introduce artificial solutions that are not valid within the established domain. It is crucial to check any derived solutions against the original problem to confirm their validity.

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