Question

What is the minimum value of \(2 \cos \theta-1 ?\) (A) ?3 (B) ?2 (C) ?1 (D) 0

Short answer

The minimum value is -3.

Step-by-step solution

Understand the range of cosine

The function \(\text{cos} \theta\) varies between \(-1\) and \(1\) inclusive. Hence, we need to consider these extremum values for \( \text{cos} \theta \).

Determine the expression at extreme values

Substitute the extreme values of \(\text{cos} \theta\) into \(2 \cos \theta - 1\). \(\begin{align*} \text{When} \ \theta = 0, \ \text{cos} \theta = 1:\ \2 \times 1 - 1 &= 2 - 1 = 1 \ \ \text{When} \ \theta = \pi, \text{cos} \theta = -1:\ 2 \times (-1) - 1 &= -2 - 1 = -3\end{align*} \)

Evaluate the minimum value

From the two extreme evaluations, it is clear that the minimum value of \(2 \cos \theta - 1\) is \(-3\).

Key concepts

cosine function range

The cosine function, represented as \(\text{cos} \theta\), is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right triangle for a given angle \(\theta\). The function oscillates between -1 and 1. This means for any angle \(\theta\), the value of \(\text{cos} \theta\) will always lie within this range.

The notation of this can be written as:

\(-1 \leq \text{cos} \theta \leq 1\).

This range is crucial when analyzing any trigonometric expression that involves cosine, like in our given problem.

minimum value determination

To determine the minimum value of a function, we typically evaluate it at the critical points within the given range. For trigonometric functions involving cosine, we check the values when \(\text{cos} \theta\) is at its maximum and minimum.

Here, the expression in the problem is \(2 \text{cos} \theta - 1\). To find its minimum value:

• First, know the extreme values of \(\text{cos} \theta\), which are -1 and 1.
• Next, substitute these extreme values into the given expression

The process breaks down as follows:

• When \(\text{cos} \theta = 1\): \(2 \times 1 - 1 = 1\)
• When \(\text{cos} \theta = -1\): \(2 \times (-1) - 1 = -2 - 1 = -3\)

From these substitutions, we see that the minimum value of the expression is -3.

extreme value substitution

Extreme value substitution involves plugging the extreme values of a variable within its range into the function to determine the highest or lowest possible outcome.

For the problem \(2 \text{cos} \theta - 1\), we use the range of \(\text{cos} \theta\), which lies between -1 and 1.

By substituting these extremes:

• When \(\text{cos} \theta = 1\): \(2 \times 1 - 1 = 1\)
• When \(\text{cos} \theta = -1\): \(2 \times (-1) - 1 = -3\)

Substitution ensures that we evaluate every potential outcome for the given range, providing clarity on the minimum and maximum values of the expression.

This method is particularly useful in trigonometric problems where functions can oscillate within a defined range.