Problem 64 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\sin t = -\frac{3}{5}\); \(\tan t = \frac{3}{4}\).

Step 1: Understand the given information

We are provided that \( \cos t = -\frac{4}{5} \), and the terminal point of \( t \) is located in the third quadrant. Knowing this, we need to find the corresponding sine and tangent values for the angle.

Step 2: Find the sine of the angle using Pythagoras Identity

In trigonometry, we can use the identity \( \sin^2 t + \cos^2 t = 1 \). Given \( \cos t = -\frac{4}{5} \), substitute to find \( \sin t \):\[\sin^2 t + \left( -\frac{4}{5} \right)^2 = 1\]\[\sin^2 t + \frac{16}{25} = 1\]\[\sin^2 t = 1 - \frac{16}{25}\]\[\sin^2 t = \frac{9}{25}\]Since the angle is in Quadrant III, \( \sin t \) is negative, so:\[\sin t = -\frac{3}{5}\]

Step 3: Calculate the tangent of the angle

Tangent is defined as \( \tan t = \frac{\sin t}{\cos t} \). Substituting our values in, we get:\[\tan t = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}\]

Step 4: Review Result

We have calculated the values as follows: \( \sin t = -\frac{3}{5} \) and \( \tan t = \frac{3}{4} \). All these values are consistent with \( t \) being in the third quadrant.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a central concept in trigonometry that relates the squares of the sine and cosine of an angle. It can be written as:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity emerges from the Pythagorean Theorem applied in the unit circle, where the radius is 1. Because of this identity, if either sine or cosine is known for a given angle, the other can be determined.
For example, if you know \( \cos t = -\frac{4}{5} \), you can find \( \sin t \) by rearranging the identity:

• \( \sin^2 t = 1 - \cos^2 t \)
• \( \sin^2 t = 1 - \left(-\frac{4}{5}\right)^2 \)

With these simple steps, the value of another trigonometric function can be easily found using the Pythagorean Identity.

Cosine Function

The cosine function, represented as \( \cos \), is a fundamental trigonometric function expressing the ratio of the adjacent side to the hypotenuse in a right triangle. Cosine also represents the x-coordinate of a point on the unit circle.
In the context of the unit circle, \( \cos t \) indicates how far along the horizontal axis (adjacent to the angle) the terminal point of angle \( t \) lies.
Given \( \cos t = -\frac{4}{5} \), this means the point is further along the negative x-axis, which aligns with the scenario in Quadrant III.

• Cosine is negative in the third and fourth quadrants.

Understanding the sign of cosine helps in determining the possible quadrants where an angle may terminate.

Sine Function

The sine function, or \( \sin \), measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. In the unit circle, \( \sin t \) is the y-coordinate of the terminal point.
In Quadrant III, both sine and cosine are negative. Thus, if \( \sin^2 t = \frac{9}{25} \), then \( \sin t = -\sqrt{\frac{9}{25}} \), which simplifies to \( -\frac{3}{5} \).

• Sine is positive in Quadrants I and II.
• Sine is negative in Quadrants III and IV.

The sine function is essential for analyzing vertical motion and oscillations in various applications. Recognizing the sign based on the quadrant can help find the correct value.

Tangent Function

The tangent function, \( \tan \), is derived as the quotient of sine and cosine functions:

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

Tangent gives the slope of the line corresponding to the angle \( t \) on the unit circle.
In our problem, since we have \( \sin t = -\frac{3}{5} \) and \( \cos t = -\frac{4}{5} \), the tangent is:

• \( \tan t = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4} \)

Tangent is positive in the third quadrant, as both sine and cosine are negative, their ratio is positive.

• Tangent is positive in Quadrants I and III.
• Tangent is negative in Quadrants II and IV.

Mastering tangent helps to assess slopes in graphs and is used extensively in calculus.

Quadrants in Trigonometry

Understanding the quadrants in trigonometry is critical for finding the correct signs of trigonometric functions. The Cartesian plane is divided into four quadrants:

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

The angle \( t \) in this problem is in Quadrant III. Here, you can deduce that both \( \sin t \) and \( \cos t \) should be negative. This affects the answers for trigonometric functions like tangent, which is positive when both inputs (sine and cosine) are negative.
This knowledge is fundamental when solving problems with angles on the unit circle and helps ensure accurate interpretation of trigonometric identities and function values.