Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-0.56$$

Step-by-Step Solution

Answer

The statement is possible for some angle \( \theta \) because -0.56 is within the range of cosine values.

Step 1: Understand the Range of Cosine Function

The cosine of any angle \( \theta \) ranges from -1 to 1. This means that for any angle \( \theta \), \( \cos \theta \) will always produce a value between -1 and 1.

Step 2: Evaluate the Given Statement

The statement is \( \cos \theta = -0.56 \). Since \(-1 \leq \cos \theta \leq 1\), the value -0.56 falls within this range. This makes the statement potentially possible for some angle \( \theta \).

Step 3: Conclusion Based on Evaluation

Since -0.56 is between the values of -1 and 1, the statement \( \cos \theta = -0.56 \) is possible for some angle \( \theta \). Any angle that satisfies this equation would make it true.

Key Concepts

Angle Theta

Angles are a fundamental part of trigonometry. Angle \( \theta \) is a way to describe the rotation between two rays that share a common endpoint, called the vertex. It is measured in degrees or radians.

• Degrees are a traditional unit where a full circle is 360 degrees.
• Radians are a more mathematical approach where a full circle is \(2\pi\) radians.

Understanding angles is crucial as they help define the trigonometric functions such as sine, cosine, and tangent. These functions relate the angles of a triangle to the lengths of its sides. For instance, in the unit circle, which is a circle with a radius of 1, the angle \(\theta\) determines the coordinates of a point on the circle's circumference.Each angle \(\theta\) can have different cosine values, and this will happen on various positions around the circle, explaining the existence of positive and negative values for cosine.

Possible Values of Cosine

The cosine function is one of the primary trigonometric functions and is widely used to calculate the horizontal position of a point on a unit circle given an angle \( \theta \). The cosine values range from -1 to 1.

• This range is deeply connected to the properties of a circle and the concept of rotation.
• Cosine represents the x-coordinate of a unit circle, which means it varies as the angle \( \theta \) sweeps around the circle.

Thus, when you calculate \( \cos \theta \), the result will always fall somewhere within that interval from -1 to 1. For example, when looking at \( \cos \theta = -0.56 \), it falls well within the acceptable range, making it a possible outcome for some \( \theta \). Knowing this range ensures that any calculation involving the cosine function respects its boundaries and does not produce impossible results.

Trigonometric Range

The trigonometric range refers to the set of all possible values that a trigonometric function, like cosine, can take. This concept is specifically about what outputs you can expect when you input different angles into these functions.

• For the cosine function, this range is always between -1 and 1.
• This is because the cosine of an angle represents the x-coordinate of a point on the unit circle, and thus cannot exceed the radius, which is 1.

Understanding trigonometric range helps you determine the feasibility of any given output. It tells you whether or not a particular value can actually result from the function. For example, when someone claims \( \cos \theta = 1.5 \), you immediately know this is impossible, as it falls outside the established range. However, a claim like \( \cos \theta = -0.56 \) is plausible, as it fits comfortably within this span. Therefore, mastering the range of trig functions allows for accurate interpretation of angles and their corresponding coordinate values.