Show that each of the following is not an identity by...

Show that each of the following is not an identity by finding a replacement or replacements for which the sides of the equation do not name the same number. Then use a graphing calculator to show that the equation is not an identity.
$$\sqrt{\sin ^{2} \theta}=\sin \theta$$

Key Concepts

Trigonometric Function Properties

Trigonometric functions like sine have specific ranges and periodic behaviors. In the context of this question, the sine function takes values between -1 and 1. Recognizing that sin? can be negative is key because it shows that ?(sin²?) equals |sin?|, which is not equal to sin? when sin? is negative. This property is integral to understanding why some trigonometric equations may not hold true for all angles.

Identity Equations

An identity is an equation that holds true for all values within the domain of the variable involved. Demonstrating that a given equation fails to be an identity usually involves finding a counterexample where the two sides of the equation yield different values. Understanding identities is crucial in verifying the universality of mathematical equations.

Counterexample Substitution

Using counterexamples involves substituting specific values into an equation to check whether both sides yield the same result. If a single counterexample is found where the equality fails, then the equation is confirmed not to be an identity. This concept is a common method to disprove universal claims in mathematics.

Non-negative Square Root

The square root function, when applied to an expression squared, results in the non-negative value of that expression. That is, ?(x²) equals |x|, not x. This fundamental property is crucial because it sets the inherent range of the square root to be non-negative, affecting any equation where a square root of a squared function appears.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, disregarding its sign. When an equation involves ?(function²), it is equivalent to the absolute value of that function. Recognizing the role of absolute value is essential to understanding why an equation like ?(sin²?) = sin? does not hold for all values of ?.

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