Determine whether each statement is possible or impossible. b. tan θ = 1.4

Verified step by step guidance

1. Recall that the tangent function, \(\tan \theta\), is defined as the ratio of the sine and cosine of an angle: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
2. Understand that the range of the tangent function is all real numbers, meaning \(\tan \theta\) can take any real value from \(-\infty\) to \(+\infty\).
3. Since \(1.4\) is a real number, check if it lies within the range of possible values for \(\tan \theta\). Because the tangent function can be any real number, \(1.4\) is within this range.
4. Conclude that there exists some angle \(\theta\) such that \(\tan \theta = 1.4\), making the statement possible.
5. Optionally, to find such an angle, you could use the inverse tangent function: \(\theta = \tan^{-1}(1.4)\), which will give an angle in radians or degrees.

Key Concepts

Definition and Range of the Tangent Function

The tangent of an angle θ in a right triangle is defined as the ratio of the opposite side to the adjacent side. Unlike sine and cosine, tangent values can range from negative to positive infinity, meaning any real number is a possible value for tan θ.

Domain and Range of Tangent on the Unit Circle

On the unit circle, tan θ is defined as sin θ divided by cos θ. Since cosine can be zero, tan θ is undefined at odd multiples of 90°, but between these points, tan θ takes all real values, including 1.4, making such values possible.

Determining Possibility of a Tangent Value

To determine if tan θ = 1.4 is possible, consider that tangent can take any real number except where cosine is zero. Since 1.4 is a real number and not restricted by the function's domain, tan θ = 1.4 is possible for some angle θ.