Problem 58 - 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

\(\tan t = \frac{\sin t}{\cos t} = \tan t\) since the tangent is positive in Quadrant III.

Step 1: Recall relationship between trigonometric functions

In any right triangle on the Cartesian plane, the tangent of an angle is given by \[\tan t = \frac{\sin t}{\cos t} \]where \(\sin t\) is the opposite side over hypotenuse, and \(\cos t\) is the adjacent side over hypotenuse.

Step 2: Consider the signs of trigonometric functions in Quadrant III

In Quadrant III, both \(\sin t\) and \(\cos t\) are negative, but since tangent is sine divided by cosine, \(\tan t\) is positive in this quadrant.

Step 3: Solve expression for \(\tan t\) in terms of \(\cos t\)

Rearrange the equation from Step 1 to solve for \(\tan t\):\[\tan t = \frac{\sin t}{\cos t}\]Since \(\tan t = \frac{1}{\cos t} \cdot \sin t\), to find \(\sin t\), rearrange to:\[\sin t = \tan t \cdot \cos t\]

Step 4: Verify expression considering signs in Quadrant III

In Quadrant III, both \(\sin t\) and \(\cos t\) are negative. Ensure that multiplying \(\tan t\) (positive) with a negative \(\cos t\) will give a negative \(\sin t\). Thus, in terms of \(\cos t\), the expression becomes:\[\tan t = \frac{\sin t}{\cos t} = \frac{-|\sin t|}{-|\cos t|} = \frac{|\sin t|}{|\cos t|}\] which confirms \(\tan t = \tan t\).

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It is defined in a right-angled triangle as the ratio of the opposite side to the adjacent side. Mathematically, this is expressed as \(\tan t = \frac{\sin t}{\cos t} \), where \( \sin t \) is the sine function representing the opposite side over hypotenuse, and \( \cos t \) is the cosine function indicated by the adjacent side over hypotenuse. This relationship shows that tangent is heavily reliant on both sine and cosine, making it essential to understand their interaction.

Occasionally, you will find the tangent function being used to determine angles or side lengths in various applications, from simple classroom exercises to complex engineering fields.
Additionally, an important aspect of tangent is that it becomes undefined where the cosine function equals zero, making its graph have repeated vertical asymptotes.

• The function is periodic, with a period of \( \pi \).
• The tangent function repeats its shape every \( \pi \) radians.

Understanding tangent's behavior will help better predict outcomes across all quadrant scenarios in the Cartesian plane.

Cosine Function

The cosine function, often denoted as \( \cos t \), is another critical element of trigonometry. As a trigonometric ratio, it represents the adjacent side over the hypotenuse in a right-angle triangle.

One notable feature of cosine is its period and symmetry. The cosine function has a periodicity of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units.

• It is an even function, satisfying the identity \( \cos(-t) = \cos t \).
• This symmetry is useful when plotting cosine waves or solving equations.

Cosine plays a crucial role not only in defining the tangent function, as seen in \( \tan t = \frac{\sin t}{\cos t} \), but also in other identity derivations. When working with trigonometric functions' signs in various quadrants, cosine provides insight into the behavior of angles and side lengths, especially in Quadrant III, where the cosine value is negative.

Quadrants in Trigonometry

Trigonometry divides the coordinate system into four quadrants. Understanding each quadrant is key when working with trigonometric identities and functions such as tangent and cosine.

Here's a quick guide:

• Quadrant I: Both sine and cosine values are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine become negative; however, tangent, being the ratio \( \frac{\sin t}{\cos t} \), is positive.
• Quadrant IV: Sine is negative, and cosine is positive.

In the exercise discussed, Quadrant III plays a pivotal role. When both sine and cosine are negative, their quotient, or tangent, results in a positive value, reflecting the specific properties of this quadrant.

By understanding the behaviors of the trigonometric functions in each quadrant, you can predict their signs and easily manipulate expressions depending on the angle \( t \). This knowledge simplifies solving trigonometric equations and enhances comprehension of angle properties throughout the Cartesian plane.