Problem 66 - Algebra and Trigonometry

Question

Find the values of the trigonometric functions of \(t\) from the given information. \(\cos t=-\frac{4}{5}, \quad\) terminal point of \(t\) is in Quadrant III

Step-by-Step Solution

Answer

The values are \( \sin t = -\frac{3}{5} \), \( \cos t = -\frac{4}{5} \), \( \tan t = \frac{3}{4} \), \( \csc t = -\frac{5}{3} \), \( \sec t = -\frac{5}{4} \), and \( \cot t = \frac{4}{3} \).

Step 1: Understand the Given Information

We are given \( \cos t = -\frac{4}{5} \) and the terminal point of \( t \) is in Quadrant III. This tells us that the cosine of angle \( t \) is negative and the angle is in the third quadrant.

Step 2: Determine the Signs of the Functions

In Quadrant III, both sine and cosine values are negative, but tangent is positive because it is the ratio of sine to cosine.

Step 3: Use Pythagorean Identity to Find \( \sin t \)

Use the identity \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \cos t = -\frac{4}{5} \) to find \( \sin t \): \[ \sin^2 t = 1 - \left( -\frac{4}{5} \right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \] Thus, \( \sin t = -\sqrt{\frac{9}{25}} = -\frac{3}{5} \) (taking the negative because we are in Quadrant III).

Step 4: Calculate \( \tan t \)

Tangent is the ratio of sine to cosine: \( \tan t = \frac{\sin t}{\cos t} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4} \).

Step 5: Find the Remaining Trigonometric Functions

Compute \( \csc t = \frac{1}{\sin t} = -\frac{5}{3} \), \( \sec t = \frac{1}{\cos t} = -\frac{5}{4} \), and \( \cot t = \frac{1}{\tan t} = \frac{4}{3} \).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It provides a relationship between the sine and cosine of an angle. The identity is expressed as:\[ \sin^2 t + \cos^2 t = 1 \]This equation helps us understand how sine and cosine values work together to describe angles. When you're given either the sine or cosine value of an angle, you can use this identity to find the missing value.
Using this identity, you can solve for the sine when the cosine is known, and vice versa. For example, if the cosine of an angle is \( -\frac{4}{5} \), you can find the sine by:

• Plugging \(-\frac{4}{5}\) into the identity
• Solving for \( \sin^2 t \)
• Taking the square root

Remember, the sine value depends on which quadrant the angle is in, as it will affect the sign of sine. In our case, the negative is chosen because the angle is in Quadrant III.

quadrants

The concept of quadrants is essential in trigonometry as it helps determine the sign and values of trigonometric functions. The coordinate plane is divided into four quadrants:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

Knowing the quadrant in which your angle's terminal side lies helps determine the sign of trigonometric functions like sine and cosine. For example, since our angle \( t \) is in Quadrant III, both \( \sin t \) and \( \cos t \) are negative, which is crucial when solving trigonometric problems. Understanding the quadrants simplifies determining these values quickly.

cosine

Cosine is one of the primary trigonometric functions and is usually abbreviated as \( \cos \). It represents the ratio between the adjacent side to the hypotenuse in a right-angle triangle. In the unit circle, cosine gives the x-coordinate of an angle's intersection on the circle.
The given problem specifies \( \cos t = -\frac{4}{5} \), indicating that we are working with an obtuse angle that has its terminal point in Quadrant III. Cosine values are only positive in the first and fourth quadrants, so the negative sign aligns with the properties of Quadrant III.
The cosine function plays a significant role when working with other trigonometric identities and determining the sine, tangent, and reciprocal functions. In this exercise, understanding the behavior of cosine in different quadrants became key to unlocking the other trigonometric values associated with angle \( t \).