Problem 158 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A trigonometric equation with an infinite number of solutions is an identity.

Step-by-Step Solution

Answer

The statement is false. A correct version might be: A trigonometric equation with solutions for all variable values within its domain is an identity.

Step 1: Understand the term trigonometric equation

A trigonometric equation is an equation that involves trigonometric functions. Solutions to such equations are the values of the variable for which the equation is true. It is possible for a trigonometric equation to have infinite solutions.

Step 2: Define an identity

An identity is an equation that holds true for all the variables where both sides of the equation are defined. In other words, an identity is true regardless of the values of variables, as far as those values are within the domain of the functions in the equation.

Step 3: Analyze the Statement

A trigonometric equation could have an infinite number of solutions, due to the periodic nature of trigonometric functions. Additionally, an identity is also an equation with an infinite number of solutions. However, not all trigonometric equations with infinite solutions are identities, as a trigonometric equation could be true for infinite values of a variable without it being true for all possible values of that variable, which is a requisite for an identity. Therefore, the statement given in the exercise is false.

Step 4: Correct the Statement

A corrected version of the statement could be as follows: 'A trigonometric equation with solutions for all variable values within its domain is an identity.'