Problem 32 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(-\frac{1}{3},-\frac{2 \sqrt{2}}{3}\right)\)

Step-by-Step Solution

Answer

\(\sin t = -\frac{2\sqrt{2}}{3}, \cos t = -\frac{1}{3}, \tan t = 2\sqrt{2}\).

Step 1: Understand the terminal point

The terminal point \( P(x, y) \) is given as \( \left(-\frac{1}{3}, -\frac{2\sqrt{2}}{3}\right) \). This point lies on the unit circle, which means the distance from the origin to the point \( P(x, y) \) is 1.

Step 2: Compute \( \sin t \) using the y-coordinate

The sine of the angle \( t \) is the y-coordinate of the terminal point. Therefore, \( \sin t = -\frac{2\sqrt{2}}{3} \).

Step 3: Compute \( \cos t \) using the x-coordinate

The cosine of the angle \( t \) is the x-coordinate of the terminal point. Therefore, \( \cos t = -\frac{1}{3} \).

Step 4: Compute \( \tan t \) using \( \sin t \) and \( \cos t \)

The tangent of the angle \( t \) is the ratio of the sine to the cosine, \( \tan t = \frac{\sin t}{\cos t} = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = 2\sqrt{2} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and is a circle with a radius of 1, centered at the origin of a coordinate plane (0, 0). This makes it useful for understanding trigonometric functions, as all points on the circle correspond to angles t (measured in radians). The most useful property of the unit circle is that the x-coordinate and y-coordinate of any point on it represent the cosine and sine of the angle, respectively.

• Radius equals 1
• Center: origin (0,0)
• Angle t corresponds to the coordinates (x, y)

Using the unit circle, it's easier to calculate and visualize the sine and cosine values for any real number angle t. This concept simplifies dealing with trigonometric functions because the Pythagorean identity \(x^2 + y^2 = 1\) always holds for any point \((x, y)\) on the circle.

Sine

The sine function represents the y-coordinate of a point on the unit circle. When you have an angle t, the sine of this angle can be directly obtained by looking at the y-coordinate of the terminal point on the unit circle. In our specific problem, the terminal point given is \((-\frac{1}{3}, -\frac{2\sqrt{2}}{3})\), meaning that:

• The sine of angle t, or \(\sin t\), is equal to \(-\frac{2\sqrt{2}}{3}\).

This clearly shows how the sine directly relates to the position of the point on the circle. Knowing the y-coordinate directly gives you the sine value of the angle measured from the positive x-axis to the line segment connecting the origin to the point.

Cosine

Cosine is another trigonometric function represented as the x-coordinate of a point on the unit circle. For a given angle t, the cosine of this angle is simply the x-value where the circle intersects the terminal line of the angle. In our case, the terminal point \((-\frac{1}{3}, -\frac{2\sqrt{2}}{3})\) gives us:

• The cosine of angle t, or \(\cos t\), is \(-\frac{1}{3}\).

By interpreting cosine as the horizontal projection along the x-axis, it becomes more straightforward to understand how this function relates to an angle. It tells us how far along the x-axis we "project" from the origin to reach the point of intersection on the circle.

Tangent

Tangent is a trigonometric function that can be thought of as the slope of the line created when connecting the origin to a point on the unit circle. It is the ratio of sine to cosine: \(\tan t = \frac{\sin t}{\cos t}\).This makes tangent particularly useful for its ability to represent how steep an angle is.In our example, with \(\sin t = -\frac{2\sqrt{2}}{3}\) and \(\cos t = -\frac{1}{3}\), the tangent calculation is:

• \(\tan t = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = 2\sqrt{2}\)

Understanding tangent in this way helps make sense of its importance in trigonometry, as it links the vertical and horizontal components and allows for the calculation of angle steepness efficiently.