Problem 56 - Algebra and Trigonometry

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \cos t ; \quad\) quadrant III

Step-by-Step Solution

Answer

\( \tan t = \frac{-\sqrt{1 - \cos^2 t}}{\cos t} \) in Quadrant III.

Step 1: Understand the Given Information

We are given that \( \tan t \) and \( \cos t \) are related functions, and the terminal point determined by \( t \) lies in Quadrant III.

Step 2: Recall Trigonometric Signs in Quadrant III

In Quadrant III, both sine and cosine are negative. Hence, \( \tan t = \frac{\sin t}{\cos t} \) will be positive because it's the ratio of two negative numbers.

Step 3: Express \( \tan t \) in Terms of \( \cos t \)

The tangent function is expressed as \( \tan t = \frac{\sin t}{\cos t} \). We are tasked to express \( \tan t \) using \( \cos t \).

Step 4: Determine \( \sin t \) Using Pythagorean Identity

Since \( \sin^2 t + \cos^2 t = 1 \), \( \sin t = -\sqrt{1 - \cos^2 t} \) holds in Quadrant III because sine is negative there.

Step 5: Formulate \( \tan t \) in Terms of \( \cos t \)

Substitute \( \sin t = -\sqrt{1 - \cos^2 t} \) into \( \tan t = \frac{\sin t}{\cos t} \). Therefore, \( \tan t = \frac{-\sqrt{1 - \cos^2 t}}{\cos t} \).

Step 6: Provide the Final Expression

Thus, the expression for \( \tan t \) in terms of \( \cos t \) for Quadrant III is \( \tan t = \frac{-\sqrt{1 - \cos^2 t}}{\cos t} \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle in a right triangle, the square of the sine of the angle plus the square of the cosine of the angle equals one. This can be expressed mathematically as:

\[ \sin^2 t + \cos^2 t = 1 \]

This identity is incredibly useful as it allows us to find one function value if we know the other. For example, if you know the value of the cosine, you can easily compute the sine. Simply rearrange the equation to solve for sine:

• \( \sin^2 t = 1 - \cos^2 t \)

Notice that when taking the square root of both sides, you have two potential values: positive or negative. The sign depends on the specific quadrant the angle is positioned in. The Pythagorean identity is crucial for transforming expressions and solving trigonometric equations.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are ratios derived from the sides of a right triangle. These functions define the relationships between the angles and sides of the triangle, which are vital for calculations in trigonometry.

Here are the basic definitions:

• \( \sin t = \text{Opposite} / \text{Hypotenuse} \)
• \( \cos t = \text{Adjacent} / \text{Hypotenuse} \)
• \( \tan t = \sin t / \cos t \)

The tangent function can be described as the ratio of the sine and cosine functions. Understanding how one function can be expressed in terms of another is essential, especially in problem-solving.

In different quadrants on the unit circle, these functions take on different signs, influencing the calculations and the application of identities like the Pythagorean identity.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Each quadrant has specific rules about the sign of the trigonometric functions:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Tangent is positive, whereas sine and cosine are negative.
• Quadrant IV: Cosine is positive, while sine and tangent are negative.

This quadrant system helps in determining the sign of a trigonometric function for a given angle. Knowing the quadrant of an angle provides immediate insight into which functions are positive and which are negative.

In the exercise, understanding that the angle is in Quadrant III, where tangent is positive, is vital. This ensures correct formulation when using identities, guaranteeing the accuracy of trigonometric expressions.