Problem 50 - Precalculus Mathematics for Calculus

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\cot \theta=\frac{1}{4}, \quad \sin \theta<0$$

Step-by-Step Solution

Answer

\( \sin \theta = -\frac{4}{\sqrt{17}}, \cos \theta = \frac{1}{\sqrt{17}} \).

Step 1: Understand Cotangent

The cotangent of an angle θ is the reciprocal of its tangent, which is the ratio of adjacent side to opposite side in a right triangle. Given that \( \cot \theta = \frac{1}{4} \), this means the adjacent side is 1 and the opposite side is 4.

Step 2: Determine the Quadrant

The condition \( \sin \theta < 0 \) implies that θ is located in either the third or fourth quadrant, as sine is negative in these quadrants.

Step 3: Use Pythagorean Theorem to Find Hypotenuse

Apply the Pythagorean theorem to find the hypotenuse \( r \). If adjacent is 1 and opposite is 4, then \( r = \sqrt{1^2 + 4^2} = \sqrt{17} \).

Step 4: Calculate Sine

Use the definition of sine \( \sin \theta = \frac{{\text{opposite}}}{r} = \frac{4}{\sqrt{17}} \). Since \( \sin \theta < 0 \), adjust the sign: \( \sin \theta = -\frac{4}{\sqrt{17}} \).

Step 5: Calculate Cosine

Find cosine using the adjacent side and hypotenuse: \( \cos \theta = \frac{1}{\sqrt{17}} \). Since cosine is positive in the fourth quadrant and negative in the third, \( \cos \theta = -\frac{1}{\sqrt{17}} \) if in the third and \( \cos \theta = \frac{1}{\sqrt{17}} \) if in the fourth. Since we have \( \cot \theta = \frac{1}{4} \), ran calculation already assumed fourth quadrant, making \( \cos \theta = \frac{1}{\sqrt{17}} \).

Step 6: Confirm with Exact Values

Verify that the calculated values satisfy the original conditions. Since \( \tan \theta = \frac{1}{4} \) is positive, this confirms θ is in the fourth quadrant, where cos is positive and sin is negative, consistent with our sine value. Cosine is confirmed as \( \frac{1}{\sqrt{17}} \) since it must be positive.

Key Concepts

Understanding Cotangent

The cotangent function is one of the six fundamental trigonometric functions. It is often abbreviated as "cot." Cotangent is particularly useful when analyzing angles in right triangles.

The cotangent of an angle \( \theta \) is the reciprocal of the tangent function:

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

For the given problem, \( \cot \theta = \frac{1}{4} \), which implies that the adjacent side is 1 when the opposite side is 4, hence forming a ratio of 1:4.

Cotangent is positive in the first and third quadrants. Since \( \sin \theta < 0 \), this suggests that \( \theta \) is not in the first quadrant, but rather in the fourth quadrant.

Exploring Sine

The sine function is another key trigonometric function. It is abbreviated as "sin." Sine helps us understand the relationship between an angle and the lengths of its sides in a right triangle.

To calculate sine:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

In this exercise, we determined the hypotenuse using the Pythagorean theorem, resulting in \( \sqrt{17} \). For \( \sin \theta < 0 \), given the opposite side as 4, we find:

• \( \sin \theta = -\frac{4}{\sqrt{17}} \)

The negative sine value indicates that \( \theta \) could be in the third or fourth quadrant, aligning with the given that sine must be negative in these areas.

Applying the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially in studies involving right triangles. It states that in a right triangle, the square of the hypotenuse (\( r \)) is equal to the sum of the squares of the other two sides, commonly known as adjacent and opposite.

The theorem is expressed as:

• \( a^2 + b^2 = c^2 \)

In this exercise, the adjacent side \( a \) is 1, and the opposite side \( b \) is 4, so:

• \( 1^2 + 4^2 = r^2 \)
• \( r = \sqrt{1 + 16} = \sqrt{17} \)

This calculation of the hypotenuse allows accurate computation of other trigonometric functions such as sine and cosine, which require this information for their formulas.