Problem 47 - Precalculus

Question

Determine whether each statement is possible or not. $$\sin \theta=-0.999$$

Step-by-Step Solution

Answer

The statement is possible.

Step 1: Understanding the Sine Function Range

The sine function, \(\sin( heta)\u2013\), maps angles to values between \(-1\) and \(1\). This is the range of the sine function, meaning any value of \(\sin( heta)\u2013\) must satisfy \(-1 \leq \sin(\theta) \leq 1\).

Step 2: Checking the Given Value

In the exercise, we have \(\sin(\theta) = -0.999\u2013\). We need to see if this value fits within the range \([-1, 1]\). Since \(-0.999\u2013\) is greater than \(-1\) and less than \(1\), it falls inside the range.

Step 3: Conclusion: Evaluate the Possibility

Since \(\sin(\theta) = -0.999\u2013\) fits within the range \([-1, 1]\), the statement is possible. \(\sin(\theta)\u2013\) can indeed be \(-0.999\u2013\), making the statement valid.

Key Concepts

Sine Function Range

The sine function is a fundamental trigonometric function that assigns a sine value to each angle, regardless of its size. This sine value is always between

• -1
• 1

When you plot the sine function on a graph, it forms a smooth, wave-like curve, called a sine wave, that oscillates between these two points. This means even if an angle is very large, the sine function will never produce a sine value less than -1 or more than 1.
Knowing this range helps us determine whether given sine values are possible. In our context, if we see a sine value, we simply check if it lies within this range to verify its validity. If it does, the sine value is possible. Otherwise, it's not feasible within the framework of the sine function.

Angle Mapping

Angle mapping refers to the relationship between angles and their sine values. Each angle corresponds to a unique point on the sine function, which maps that angle to a numerical value, following the rules of trigonometry. The sine function's cyclical nature means that any attempt to map an angle involves understanding that:

• Angles can be positive or negative
• Sine values may repeat for angles that differ by multiples of full rotations (360 degrees or 2π radians)

This feature showcases how a single sine value, such as a proposed given sine of -0.999 , could match different angles. Understanding this mapping allows you to appreciate the periodic nature of sine function and the multiple possible angles that produce the same sine value.

Validity of Trigonometric Expressions

To determine the validity of a trigonometric expression means ensuring it conforms to the properties of trigonometric functions. Given an expression for sine, we check if fits within the sine function's range. This simple measure can quickly validate whether a statement is trigonometric possible. The expression \(\sin(θ) = -0.999\) is an example of a valid expression. To judge validity, remember:

• Sine values should always lie between -1 and 1
• The context and constraints of angles must be respected

Even deep into trigonometric exploration, maintaining awareness of these basics ensures that expressions you encounter or use are suitable and correct. By consistently verifying this, you'll avoid errors and gain deeper understanding of trigonometric concepts.