Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\cot \theta=-\frac{\sqrt{3}}{3}$$

Step-by-Step Solution

Answer

The values of \( \theta \) are \( 150^{\circ} \) and \( 330^{\circ} \).

Step 1: Understanding cotangent

The cotangent function is the reciprocal of the tangent function. Hence, if \( \cot \theta = -\frac{\sqrt{3}}{3} \), then we can write \( \tan \theta = -3\sqrt{3} \). In a right triangle, tangent is expressed as \( \frac{\text{opposite}}{\text{adjacent}} \). For tangent values, \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \) which is equal to \( \frac{\sqrt{3}}{3} \). Thus, we want to find an angle whose tangent is \( -\frac{1}{\sqrt{3}} \).

Step 2: Determine Reference Angle

The reference angle for which \( \tan \theta = \frac{1}{\sqrt{3}} \) is \( 30^{\circ} \), because \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \). We already know \( \cot \theta \) is negative. We must find the angles \( \theta \) where the tangent function is negative.

Step 3: Determine Quadrants

The tangent function is negative in the second and fourth quadrants. For the second quadrant, the angle can be found as \( 180^{\circ} - 30^{\circ} = 150^{\circ} \). In the fourth quadrant, the angle is \( 360^{\circ} - 30^{\circ} = 330^{\circ} \).

Step 4: Solution Conclusion

Thus, the values of \( \theta \) that satisfy \( \cot \theta = -\frac{\sqrt{3}}{3} \) within the given interval are \( 150^{\circ} \) and \( 330^{\circ} \).

Key Concepts

Cotangent function

The cotangent function is a fundamental trigonometric function, often abbreviated as \( \cot \). It is defined as the reciprocal of the tangent function. In mathematical terms, \( \cot \theta = \frac{1}{\tan \theta} \).

To understand cotangent in the context of a right triangle, think of the tangent function, which represents the ratio of the length of the side opposite the angle to the length of the side adjacent. Since the cotangent is the reciprocal of this ratio, it can be expressed as:

• \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)

In our exercise, we want to find \( \theta \) such that \( \cot \theta = -\frac{\sqrt{3}}{3} \). This tells us that the tangent of \( \theta \) will be the reciprocal of \(-\frac{\sqrt{3}}{3}\), which simplifies to:

• \( \tan \theta = -3\sqrt{3} \)

Beyond right triangles, the cotangent function is vital for analyzing wave patterns in the unit circle, particularly useful in understanding periodic and oscillatory systems, such as sound waves. It becomes negative depending on the angle's position relative to the axes.

Tangent function

The tangent function, denoted by \( \tan \), is a key function in trigonometry that you will often encounter together with sine and cosine.

The tangent of an angle \( \theta \) is defined as the ratio of the sine to the cosine of that angle:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

On a right triangle, \( \tan \theta \) can be visualized as:

• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

In our problem, we encounter the tangent in its reciprocal form as the cotangent, but realize that for the cotangent expression \( \cot \theta = -\frac{\sqrt{3}}{3} \), the related tangent is \(-3\sqrt{3}\), which guides us to seek reference angles.

The sign of the tangent function changes depending on the quadrant:

• Positive in the first and third quadrants.
• Negative in the second and fourth quadrants.

This sign change is crucial to determine in which quadrants other angles with the same reference angle could occur, maintaining the desired negative tangent value.

Reference angles

Reference angles are widely used in trigonometry to simplify the analysis of angles on the unit circle. A reference angle is the smallest angle that an angle makes with the x-axis.

To find the actual angle using its reference angle, consider its position in the coordinate plane:

• First quadrant: The reference angle is simply \( \theta \).
• Second quadrant: \( 180^{\circ} - \theta \)
• Third quadrant: \( \theta - 180^{\circ} \)
• Fourth quadrant: \( 360^{\circ} - \theta \)

In this exercise, since we are dealing with \( \tan \theta = \frac{1}{\sqrt{3}}\), the reference angle is \( 30^{\circ} \), as the tangent of \( 30^{\circ} \) is \( \frac{1}{\sqrt{3}} \).

Considering the negative tangent value, we determine that this reference angle can be reflected into quadrants where the tangent is negative: the second and fourth quadrants. Hence, the angles \( \theta = 150^{\circ} \) and \( 330^{\circ} \) meet the condition of having a tangent resulting in \(-\frac{1}{\sqrt{3}} \). Understanding reference angles helps simplify the calculation of angles, especially within specific intervals.