Determine whether each statement is possible or impossible. See Example 4. sin θ = 3

Verified step by step guidance

1. Recall the range of the sine function: for any angle \( \theta \), \( \sin \theta \) must satisfy \( -1 \leq \sin \theta \leq 1 \).
2. Examine the given value \( \sin \theta = 3 \) and compare it to the possible range of sine values.
3. Since 3 is greater than 1, it lies outside the range of possible sine values.
4. Therefore, it is impossible for \( \sin \theta \) to equal 3 for any real angle \( \theta \).
5. Conclude that the statement \( \sin \theta = 3 \) is impossible.

Key Concepts

Range of the Sine Function

The sine function outputs values only within the range of -1 to 1 for all real angles θ. This means sin θ cannot be greater than 1 or less than -1, which is crucial for determining the possibility of a given sine value.

Definition of the Sine Function

Sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Since the hypotenuse is the longest side, this ratio must always be between -1 and 1.

Evaluating Trigonometric Statements

To assess whether a trigonometric statement is possible, one must check if the given value lies within the function's range and adheres to its properties. For sin θ = 3, since 3 is outside the range [-1,1], the statement is impossible.