Problem 66 - Algebra and Trigonometry

Question

Find the values of the trigonometric functions of \(t\) from the given information. \(\tan t=\frac{1}{4},\) terminal point of \(t\) is in quadrant III

Step-by-Step Solution

Answer

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

Step 1: Analyzing the Tan Function

Given that \( \tan t = \frac{1}{4} \). This means we have the ratio \( \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{4} \). In quadrant III, both sine and cosine are negative, and thus \( \tan t \) is positive.

Step 2: Finding the Hypotenuse using Pythagorean identity

Use the Pythagorean identity to find the hypotenuse: \( r = \sqrt{x^2 + y^2} \) with \( x = -4 \), \( y = -1 \). So, \( r = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \).

Step 3: Calculate Sine Function

Using \( \sin t = \frac{y}{r} \), substitute \( y = -1 \) and \( r = \sqrt{17} \), so \( \sin t = \frac{-1}{\sqrt{17}} \).

Step 4: Calculate Cosine Function

Using \( \cos t = \frac{x}{r} \), substitute \( x = -4 \) and \( r = \sqrt{17} \), so \( \cos t = \frac{-4}{\sqrt{17}} \).

Step 5: Verify with Tan Function

Check if the values satisfy \( \tan t = \frac{\sin t}{\cos t} \). Compute \( \tan t = \frac{-1/\sqrt{17}}{-4/\sqrt{17}} = \frac{1}{4} \), confirming that values are correct.

Key Concepts

Quadrant III

Understanding the placement of an angle in the unit circle is crucial for identifying the signs of trigonometric functions. Quadrant III, which is essential in our exercise, consists of angles ranging from 180° to 270° (or \pi \ to \frac{3\pi}{2}\ radians). In this quadrant, both the sine and cosine functions are negative. This negative sign stems from the locations of the 'x' and 'y' values in the coordinate plane. Specifically, in Quadrant III:

• The x-coordinates, which relate to cosine, are negative because they lie to the left of the y-axis.
• The y-coordinates, which relate to sine, are also negative since they fall below the x-axis.

Since tangent is the ratio of sine to cosine, \tan t\ becomes positive in this quadrant (as a negative divided by a negative is positive), which aligns with the given information that \( \tan t = \frac{1}{4}\). Understanding the nature of Quadrant III helps us correctly determine the signs of sine, cosine, and tangent for given angles.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that offers a neat way to connect the sine, cosine, and tangent functions. It states that for a given angle \(t\), the square of the sine added to the square of the cosine equals one:

• \( \sin^2 t + \cos^2 t = 1 \)

To find the hypotenuse, or 'r', in trigonometric terms, we can rewrite the Pythagorean identity to align with the coordinates \(x\) and \(y\) of a point on a circle:

• \( r = \sqrt{x^2 + y^2} \)

This expression serves to translate vector components into scalar magnitudes \( r\) or the radius. In our specific exercise, we used this identity to calculate the hypotenuse when given \(x = -4\) and \(y = -1\). By plugging these into the formula, we found:

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

This hypotenuse helps calculate the sine and cosine, critical for solving trigonometric functions accurately when given limited information.

Tangent

Tangent is one of the primary trigonometric functions, and it relates the opposite side of a right triangle to the adjacent side. In the context of our unit circle, it's defined as \( \tan t = \frac{y}{x} \). One of the interesting properties of tangent is that it can be derived from the ratio of sine to cosine:

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

In our exercise, we started knowing \( \tan t = \frac{1}{4} \). This ratio gave us a vital hint about the relationships between the triangle's sides. With this information, and knowing the characteristics of Quadrant III where \(x\) and \(y\) are both negative, we determined:

• \( x = -4 \)
• \( y = -1 \)

Confirming the tangent value by verifying \( \tan t = \frac{-1/\sqrt{17}}{-4/\sqrt{17}} \) supports that the trigonometric values for this angle are valid. Thus, tangent not only serves as an angle measurement but also provides insight into the proportionality among the triangle's sides, facilitating the calculation of other trigonometric functions.