Problem 35 - Precalculus Mathematics for Calculus

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(-\frac{20}{29}, \frac{21}{29}\right)$$

Step-by-Step Solution

Answer

\(\sin t = \frac{21}{29}\), \(\cos t = -\frac{20}{29}\), \(\tan t = -\frac{21}{20}\).

Step 1: Understanding the Point on the Unit Circle

Given the terminal point \( P(x, y) = \left(-\frac{20}{29}, \frac{21}{29}\right) \), it lies on the unit circle with radius 1. This means that \( x^2 + y^2 = 1 \). We need to find \( \sin t \), \( \cos t \), and \( \tan t \) for this angle \( t \).

Step 2: Identify Cosine and Sine Values

Since \( P(x, y) \) is on the unit circle, \( \cos t = x \) and \( \sin t = y \). Therefore, \( \cos t = -\frac{20}{29} \) and \( \sin t = \frac{21}{29} \).

Step 3: Calculate Tangent

Tangent is the ratio of sine to cosine. Compute it using \( \tan t = \frac{\sin t}{\cos t} = \frac{\frac{21}{29}}{-\frac{20}{29}} \). Simplifying, we get \( \tan t = -\frac{21}{20} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of circles, waves, and oscillations. There are three primary trigonometric functions that we often use: sine ( \( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions express important relationships such as the angle's effect on a right triangle's side lengths or positions on the unit circle.

- The sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
- The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
- The tangent of an angle is the ratio of the sine of that angle to the cosine of that angle.

These relationships also translate directly to the unit circle, an essential tool in mathematics. The unit circle simplifies those definitions with a fixed radius of 1, making calculations simpler, especially when dealing with angles.

Sine and Cosine

In the context of the unit circle, the sine and cosine functions are particularly intuitive. The x-coordinate of a point on the unit circle is the cosine of the angle formed by the point and the center of the circle, while the y-coordinate is the sine of the angle. This makes it very easy to determine these values once you have a point on the unit circle.

For instance, given the terminal point \( P(x, y) = \left(-\frac{20}{29}, \frac{21}{29}\right) \), we directly associate:

• \( \cos t = x = -\frac{20}{29} \)
• \( \sin t = y = \frac{21}{29} \)

These values are derived from their definitions on the unit circle.

Moreover, due to the unit circle's property \( x^2 + y^2 = 1 \), you can always verify the correctness of given points by checking if their square values sum to 1. This reinforcement helps gain confidence in determining sine and cosine values when given these coordinates.

Tangent Calculation

The tangent function can be understood and derived as the ratio of sine to cosine. This makes it quite straightforward to calculate once you have the sine and cosine values.In our current example, we have the point on the unit circle with:

• \( \sin t = \frac{21}{29} \)
• \( \cos t = -\frac{20}{29} \)

To find the tangent, apply the formula \( \tan t = \frac{\sin t}{\cos t} \).

Substituting the values, we get:\[ \tan t = \frac{\frac{21}{29}}{-\frac{20}{29}} \]Simplifying the fractions:\[ \tan t = -\frac{21}{20} \]This indicates that the tangent of the angle is negative, which is consistent with the unit circle where angles in the second quadrant have negative tangent values.

Understanding the connection between sine, cosine, and tangent not only helps with purely mathematical problems but also has vast applications in fields ranging from engineering to physics.