Problem 48 - Algebra and Trigonometry

Question

43–50 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

In the fourth quadrant, \(\sin \theta = \frac{-4}{\sqrt{17}}\), \(\cos \theta = \frac{1}{\sqrt{17}}\), and \(\tan \theta = 4\).

Step 1: Understand the Given Information

We are given that \(\cot \theta = \frac{1}{4}\) and \(\sin \theta < 0\). The cotangent function is the reciprocal of the tangent, meaning \(\cot \theta = \frac{1}{\tan \theta}\). The fact that \(\sin \theta < 0\) indicates that \(\theta\) is in either the third or fourth quadrant, where sine values are negative.

Step 2: Determine the Quadrant

Since \(\cot \theta\) is positive and \(\sin \theta < 0\), \(\theta\) must be in the fourth quadrant. This is because cotangent (a ratio of adjacent/opposite) is positive where tangent is positive (first and third quadrants) and sine is negative (third and fourth quadrants). Hence, the overlap in conditions is satisfied only in the fourth quadrant.

Step 3: Use Cotangent to Find the Triangle's Sides

From \(\cot \theta = \frac{1}{4}\), we can represent this as \(\text{adjacent side}/\text{opposite side} = \frac{1}{4}\). Assume the opposite side \(y = 4k\) and adjacent side \(x = 1k\), where \(k\) is a scaling factor.

Step 4: Calculate the Hypotenuse

Using the Pythagorean theorem, find the hypotenuse \(r\): \[ \sqrt{x^2 + y^2} = \sqrt{(1k)^2 + (4k)^2} = \sqrt{1k^2 + 16k^2} = \sqrt{17k^2} = \sqrt{17}k. \]

Step 5: Calculate the Sine

Since \(y = -4k\) (because \(\sin \theta < 0\)), the sine function is \ \(\sin \theta = \frac{-4k}{\sqrt{17}k} = \frac{-4}{\sqrt{17}}\). Ensure the negative sign to reflect \(\sin \theta < 0\).

Step 6: Calculate the Cosine

The cosine function is \ \(\cos \theta = \frac{x}{r} = \frac{1k}{\sqrt{17}k} = \frac{1}{\sqrt{17}}\).

Step 7: Calculate the Tangent

The tangent function is the reciprocal of the cotangent \ \(\tan \theta = \frac{1}{\cot \theta} = 4\).

Step 8: Calculate the Secant

The secant function is the reciprocal of the cosine \ \(\sec \theta = \frac{\sqrt{17}}{1} = \sqrt{17}\).

Step 9: Calculate the Cosecant

The cosecant function is the reciprocal of the sine \ \(\csc \theta = \frac{\sqrt{17}}{-4}\).

Key Concepts

Cotangent in Trigonometry

Cotangent is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. This means that \( \cot \theta = \frac{1}{\tan \theta} \).
In a right triangle, cotangent of an angle \( \theta \) is calculated as the ratio of the length of the adjacent side to the length of the opposite side.
Essentially, it can be expressed as:\[ \cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} \]

Key Points about Cotangent:

• It is undefined for angles where tangent is zero.
• It repeats its values every 180° (or \(\pi\) radians).
• It is positive in the first and third quadrants, and negative in the second and fourth quadrants.

Understanding the Sine Function

The sine function is another fundamental trigonometric function. It is defined for angles in a right triangle. The sine of an angle \( \theta \) is the ratio of the opposite side to the hypotenuse.

Consider a right triangle:

• The sine function is calculated as \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)
• Values of sine range from -1 to 1.

The sign of the sine function can help determine which quadrant the angle lies in:

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

The fact that \( \sin \theta < 0 \) clearly indicates that the angle is located in the third or fourth quadrant.

Significance of Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four regions known as quadrants, which help determine the signs of trigonometric functions:

1. **First Quadrant**: All trigonometric functions are positive.
2. **Second Quadrant**: Sine is positive, while cosine and tangent are negative.
3. **Third Quadrant**: Tangent is positive, while sine and cosine are negative.
4. **Fourth Quadrant**: Cosine is positive, while sine and tangent are negative.

Given the problem conditions \( \cot \theta = \frac{1}{4} \) (positive) and \( \sin \theta < 0 \), the angle \( \theta \) must be situated in the fourth quadrant, where cotangent is positive, and sine is negative.

Applying the Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially concerning right triangles. It states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. In equation form, it looks like:

\[ a^2 + b^2 = c^2 \]

For our trigonometry problem, if we denote the adjacent side as \( x \) and the opposite side as \( y \), then the hypotenuse \( r \) is calculated as:
\[ r = \sqrt{x^2 + y^2} \]

In the exercise, applying the Pythagorean theorem allowed us to solve for the hypotenuse (\( \sqrt{17}k \)), given that\( \cot \theta = \frac{1}{4} \), with the adjustments to the opposite and adjacent sides being made to fit this ratio.