Problem 35 - Precalculus

Question

Find the indicated trigonometric function values. If \(\cot \theta=-\sqrt{3}\) and the terminal side of \(\theta\) lies in quadrant \(I V,\) find \(\sec \theta\)

Step-by-Step Solution

Answer

\( \sec \theta = 2 \).

Step 1: Identify the Value from Cotangent

The cotangent of the angle, given as \( \cot \theta = -\sqrt{3} \), suggests that \( \tan \theta = -\frac{1}{\sqrt{3}} \). Since tangent equals sine divided by cosine, we determine that the reference angle \( \theta = \frac{\pi}{6} \) or 30 degrees.

Step 2: Determine Sign of Cosine in Quadrant IV

In quadrant IV, the cosine function is positive. Since secant is the reciprocal of cosine, it will also be positive.

Step 3: Use Identity to Find Secant

We use the identity \( \sec \theta = \frac{1}{\cos \theta} \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{1}{\sqrt{3}} \), we find \( \cos \theta = \frac{1}{2} \). Thus, \( \sec \theta = \frac{1}{(\cos \theta)} = \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot \theta \), is the reciprocal of the tangent function. This means that \( \cot \theta = \frac{1}{\tan \theta} \). Since tangent is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, cotangent becomes the ratio of the adjacent side to the opposite side.
In simpler terms, if you know the tangent of an angle, you can find the cotangent by taking the reciprocal of the tangent value. For instance, when \( \cot \theta = -\sqrt{3} \), it implies that \( \tan \theta = -\frac{1}{\sqrt{3}} \).
Understanding the relationship between tangent and cotangent helps solve trigonometric problems by leveraging known values.

• \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)
• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

Secant

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function, which means \( \sec \theta = \frac{1}{\cos \theta} \). Cosine, as known from trigonometry, is the ratio of the adjacent side to the hypotenuse in a right triangle. Therefore, secant represents the ratio of the hypotenuse to the adjacent side.
To find secant, one must first determine the cosine of the angle in question. In situations where you have the value of \( \tan \theta \), and you are aware of which quadrant the angle lies in, calculating secant requires using the identity \( \sec \theta = \frac{1}{\cos \theta} \).

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \)

For \( \cot \theta = -\sqrt{3} \) in the first provided step, we determined that \( \cos \theta = \frac{1}{2} \), which gives \( \sec \theta = 2 \) in Quadrant IV where cosine is positive.

Quadrant Analysis

Quadrant analysis is crucial in determining the signs and values of trigonometric functions. The standard position of an angle is one whose vertex is at the origin of the coordinate plane with one side lying along the positive x-axis. When the angle's terminal side ends in one of the four quadrants, we use quadrant analysis to determine the sign of trigonometric functions.
Each quadrant affects the signs:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine and cosecant are positive, others are negative.
• Third Quadrant: Tangent and cotangent are positive, others are negative.
• Fourth Quadrant: Cosine and secant are positive, others are negative.

In the exercise, knowing \( \theta \) is in the Fourth Quadrant means \( \cos \theta \) and \( \sec \theta \) are positive even if \( \cot \theta \) was negative.

Trigonometric Identities

Trigonometric identities are fundamental relationships among the trigonometric functions. Using these identities allows solving complex trigonometric problems by simplifying expressions. Some of the most common identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities, such as \( \sec \theta = \frac{1}{\cos \theta}\) and \( \cot \theta = \frac{1}{\tan \theta} \).
• Quotient identities, like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In our given problem, we applied the reciprocal identity \( \sec \theta = \frac{1}{\cos \theta} \) and used knowledge of the tangent's relationship to sine and cosine to find \( \cos \theta \) and subsequently \( \sec \theta \). This process highlights how mastering trigonometric identities simplifies evaluating various trigonometric functions.