Problem 38 - Contemporary Precalculus

Question

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

Step-by-Step Solution

Answer

Answer: The value of $$\tan t$$ is $$\frac{\sqrt{21}}{2}$$.

Step 1: Determine the Quadrant

Based on the given angle range, $$\pi < t < \frac{3\pi}{2}$$, we can determine that angle t is in the third quadrant.

Step 2: Calculate $$\sin t$$ using Pythagorean Identity

Since $$\cos t = -\frac{2}{5}$$, and t is in the third quadrant where both sine and cosine have negative values, we can use the Pythagorean Identity to find $$\sin t$$. The identity states: $$\sin^{2}t + \cos^{2}t = 1$$ Substitute the given $$\cos t$$ and solve for $$\sin t$$: $$\sin^{2}t + \left(-\frac{2}{5}\right)^{2} = 1$$ $$\sin^{2}t + \frac{4}{25} = 1$$ $$\sin^{2}t = 1 - \frac{4}{25}$$ $$\sin^{2}t = \frac{21}{25}$$ $$\sin t = \pm\sqrt{\frac{21}{25}}$$ Since t is in the third quadrant (where sine is negative): $$\sin t = -\sqrt{\frac{21}{25}}$$

Step 3: Calculate $$\tan t$$ using the definition of tangent

Now, we will use the definition of tangent, which is: $$\tan t = \frac{\sin t}{\cos t}$$ Substitute the known values of $$\sin t$$ and $$\cos t$$: $$\tan t = \frac{-\sqrt{\frac{21}{25}}}{-\frac{2}{5}}$$ Simplify the expression: $$\tan t = \frac{\sqrt{21}}{2}$$ So, the value of $$\tan t$$ is $$\frac{\sqrt{21}}{2}$$.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine of an angle. It is given by the equation \( \sin^2 t + \cos^2 t = 1 \). This identity is incredibly useful because it allows you to solve for one trigonometric function if you know the other. It draws its name from the famous Pythagorean Theorem, which relates to the lengths of the sides of a right triangle. In the context of trigonometry, this identity helps in finding unknown trigonometric functions, such as when we know \( \cos t = -\frac{2}{5} \) in the third quadrant. Knowing that sine is also negative in this quadrant, we use the identity to find \( \sin t \) by rearranging the formula to:

• Mention \( \sin^2 t = 1 - \cos^2 t \)
• Replace \( \cos t \) with its value, square the cosine value, then subtract from 1 to find \( \sin^2 t \)

Finally, because we are in the third quadrant where the sine is negative, we take the negative square root to find \( \sin t \). This demonstrates the utility of the Pythagorean Identity in solving trigonometric problems.

Unit Circle

The Unit Circle is a crucial tool in understanding trigonometric functions. It's a circle with a radius of 1 centered at the origin of a coordinate plane. The Unit Circle correlates angles, measured in radians, with points on the circle, which represent the sine and cosine values of those angles.When angles are placed on this circle:

• The x-coordinate of a point is \( \cos t \)
• The y-coordinate is \( \sin t \)

For angles between \( \pi \) and \( \frac{3\pi}{2} \), we find ourselves in the third quadrant. In this quadrant, both the sine and cosine values are negative.The Unit Circle simplifies complex calculations and visualizes trigonometric ratios. It clarifies why for \( t \) in the third quadrant, \( \sin t \) is negative, helping solve \( \tan t \) calculated as \( \frac{\sin t}{\cos t} \). Using the understanding of coordinates as trigonometric values offers a robust perspective for solving problems.

Trigonometric Ratios

In trigonometry, understanding the three primary trigonometric ratios—sine, cosine, and tangent—is essential. These ratios relate the angles of a right triangle to its side lengths, and they extend to the unit circle to correspond with angle measures.The tangent of an angle, \( t \), is defined as the ratio of its sine to its cosine: \[\tan t = \frac{\sin t}{\cos t}\]Knowing this relationship is vital for correctly solving angles like in the problem where \( t \) is in the third quadrant. These ratios enable calculations involving angles and distances and are critical in various applications, such as physics and engineering.In practical terms, once the sine and cosine values are known—\( \cos t = -\frac{2}{5} \) and \( \sin t = -\sqrt{\frac{21}{25}} \)—tangent can be computed easily. Simplifying this gives us \( \frac{\sqrt{21}}{2} \) for \( \tan t \), knowing that both sine and cosine are negative, hence the tangent is positive in this quadrant.Trigonometric ratios are not just confined to solving geometry problems but are foundational in understanding waveforms, oscillations, and even sound.