Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-0.96$$

Step-by-Step Solution

Answer

The statement \(\cos \theta = -0.96\) is possible.

Step 1: Understanding the range of cosine

The cosine of an angle in a right triangle represents the adjacent side divided by the hypotenuse, and for a unit circle, cos(θ) corresponds to the x-coordinate of the point on the circle. The range of the cosine function for all angles is from -1 to 1 inclusive, which can be expressed as \(-1 \leq \cos \theta \leq 1\).

Step 2: Analyzing the given statement

The given statement asks if \( \cos \theta = -0.96 \) is possible for some angle \( \theta \). We need to determine if this value falls within the range identified in Step 1. Since \(-1 \leq -0.96 \leq 1\), it fits within the range of possible cosine values.

Step 3: Conclusion based on the analysis

Since \( -0.96 \) is within the range of possible values for the cosine of an angle, it is indeed possible for \( \cos \theta = -0.96 \) for some angle \( \theta \). Therefore, the statement is possible.

Key Concepts

Range of Cosine

The cosine function is one of the fundamental trigonometric functions used in mathematics, particularly in geometry and calculus. When we talk about the range of the cosine function, we are referring to the set of possible output values it can produce.
For any given angle \( \theta \), the cosine function provides an output between -1 and 1. This means that if you were to plot the cosine function on a coordinate plane, its graph would oscillate in a wave form, never going above 1 or below -1.
Therefore, whenever you see a statement like \( \cos \theta = -0.96 \), you can validate its possibility by checking if \(-1 \leq -0.96 \leq 1\). Because -0.96 is indeed between -1 and 1, it falls into the possible range of cosine, making the statement valid.

Trigonometric Functions

Trigonometric functions are essential tools in mathematics for analyzing triangles, especially right-angled ones. They relate the angles of a triangle to the ratios of its sides, providing a bridge between geometry and algebra.
The primary trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function has its own specific role:

• Sine relates the opposite side of a right triangle to its hypotenuse.
• Cosine relates the adjacent side to its hypotenuse.
• Tangent relates the opposite side to the adjacent side.

It's vital to understand these basics because they allow one to determine unknown side lengths or angles in geometric problems. Furthermore, trigonometric functions form the foundation for more advanced mathematical concepts, such as Fourier transforms and wave analysis.

Unit Circle

The unit circle is a crucial concept in trigonometry that helps visualize and define trigonometric functions beyond right triangles. It is a circle with a radius of one unit, centered at the origin of a coordinate plane.
On the unit circle, each angle \( \theta \) corresponds to a specific point \((x, y)\) on the circle, where the cosine of the angle is the x-coordinate and the sine is the y-coordinate. Thus, for any angle \( \theta \), we have:

• \( \cos \theta = x \), and
• \( \sin \theta = y \)

This approach allows us to extend trigonometric definitions to any angle, even those greater than 90 degrees, using their coordinates on the unit circle. By observing the unit circle, students can better understand the periodic nature of trigonometric functions and how these functions behave across different quadrants.