Problem 1 - Algebra 2 and Trigonometry

Question

Explain why sec \(\theta\) cannot equal 0.5

Step-by-Step Solution

Answer

Secant \( \sec \theta \) cannot be 0.5 because it implies \( \cos \theta = 2 \), which is outside the range \([-1, 1]\) of cosine.

Step 1: Understanding Secant

The secant of an angle \( \theta \) is defined as \( \sec \theta = \frac{1}{\cos \theta } \). This means that \( \sec \theta \) is the reciprocal of the cosine function.

Step 2: Relationship with Cosine.

Since \( \sec \theta = \frac{1}{\cos \theta } \), if \( \sec \theta = 0.5 \), then \( \frac{1}{\cos \theta} = 0.5 \), which implies that \( \cos \theta = 2 \).

Step 3: Range of Cosine Function

The cosine function \( \cos \theta \) has a range of values between -1 and 1 inclusive, meaning \( -1 \leq \cos \theta \leq 1 \). Therefore, it is impossible for \( \cos \theta \) to equal 2, as it exceeds the allowed range.

Step 4: Conclusion

Since \( \cos \theta \) cannot be 2, \( \sec \theta \) cannot be 0.5. This conclusion is based on the relationship between cosine and secant and the range limitations of the cosine function.

Key Concepts

Secant Function

In trigonometry, the secant function is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function.
To put it simply: if you have an angle \( \theta \), the secant of this angle is denoted as \( \sec \theta = \frac{1}{\cos \theta} \).
This relationship is used especially in calculus and geometry when dealing with angles and right triangles.

• Secant is useful for angles beyond the right-angled triangle definitions.
• It helps in extending trigonometric analysis around the unit circle.
• It's vital for solving equations where cosine reaches values close to zero.

Understanding secant as the inverse of the cosine allows us to explore trigonometric identities and equations deeper, offering a clearer insight into the behavior of angles and length ratios.

Cosine Function

The cosine function is fundamental to trigonometry. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle when looking at an angle \( \theta \). Mathematically, it is expressed as \( \cos \theta \).
This function is periodic, meaning it repeats its values in regular intervals.

• The cosine function has a period of \(2\pi\). This means that the values of \( \cos \theta \) repeat every \(2\pi\) radians.
• Its graph is a smooth curve, alternating between -1 and 1.
• The highest point or maximum value of the curve is 1, and the lowest or minimum value is -1.

This behavior of the cosine function is essential for understanding why certain values, such as \(\cos \theta = 2\), are impossible since they lie outside its range.

Range of Trigonometric Functions

The concept of range is crucial when discussing trigonometric functions. For any function, the range is the set of all possible output values it can produce. For trigonometric functions, understanding their range helps determine which values are achievable and which are not.
In the case of the cosine function, its range is between -1 and 1 inclusive.

• This range means that \(-1 \leq \cos \theta \leq 1\) for any angle \(\theta\).
• The restricted range of cosine dictates the behavior of its reciprocal, the secant.
• As \(\cos \theta\) approaches the boundaries of its range, the secant function, \(\sec \theta\), approaches infinity or zero.

Thus, claims like \(\sec \theta = 0.5\) or equivalently \(\cos \theta = 2\) are impossible under normal trigonometric conditions, as they exceed the specified output limits of the cosine function's range.