Problem 30 - 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 \(P\) is \((-\frac{1}{3}, -\frac{2\sqrt{2}}{3})\). This point lies on the unit circle where the x-coordinate represents \(\cos t\) and the y-coordinate represents \(\sin t\).

Step 2: Find Sine Function

To find \(\sin t\), identify the y-coordinate of the terminal point. Here, \(\sin t = -\frac{2\sqrt{2}}{3}\).

Step 3: Find Cosine Function

To find \(\cos t\), identify the x-coordinate of the terminal point. Here, \(\cos t = -\frac{1}{3}\).

Step 4: Calculate Tangent Function

The tangent of angle \(t\) is defined as \(\tan t = \frac{\sin t}{\cos t}\). Substitute the known values: \(\tan t = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}}\). Simplify to get \(\tan t = 2\sqrt{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin in the coordinate plane. Every point on the unit circle corresponds to an angle, and these points can be described using coordinates \((x, y)\). In the context of trigonometry, the x-coordinate of a point on the unit circle represents the cosine value of the angle, while the y-coordinate symbolizes the sine value.

• The circle helps explain fundamental trigonometric functions and identities.
• Positions on the circle translate angles to functions based on the circle's properties.
• This means for any angle \(t\), the point \((\cos t, \sin t)\) lies on the unit circle.

When given a terminal point, such as \(P(x, y) = \left(-\frac{1}{3}, -\frac{2\sqrt{2}}{3}\right)\), we can immediately determine \(\cos t = -\frac{1}{3}\) and \(\sin t = -\frac{2\sqrt{2}}{3}\). Recognizing these coordinates helps simplify finding trigonometric values through geometric context.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate an angle in a right triangle to the ratio of different sides. In the circle approach, they define the coordinates of points on the unit circle.

• Sine (\(\sin t\)): Relates to the y-coordinate on the unit circle. For the terminal point given as \((-\frac{1}{3}, -\frac{2\sqrt{2}}{3})\), the sine value of angle \(t\) is \(-\frac{2\sqrt{2}}{3}\).
• Cosine (\(\cos t\)): Corresponds to the x-coordinate on the unit circle. In the exercise, \(\cos t = -\frac{1}{3}\).

These functions are periodic, meaning they repeat their sequences over intervals. Typically, the standard interval used to describe these cycles is \([0, 2\pi]\) for radians or \([0°, 360°]\) for degrees. With a unit circle providing a steady reference, angles and their trigonometric results can be swiftly figured through the position of points.

Tangent

Tangent is another crucial trigonometric function and it is the quotient of sine and cosine for a particular angle \(t\). In a right triangle, this represents the ratio between the opposite and adjacent sides to the angle.
For unit circle calculations, \(\tan t\) can be determined as \(\tan t = \frac{\sin t}{\cos t}\). Thus, using the given point:\[\tan t = \frac{-\frac{2\sqrt{2}}{3}}{-\frac{1}{3}}\]Simplifying this fraction yields \(\tan t = 2\sqrt{2}\).

• The tangent function can dictate both orientation and slopes of line segments.
• Understanding its calculated result offers insight into the properties and relationships between trigonometric segments and various geometric figures.

Remember, while sine and cosine values are limited to the range of -1 to 1 due to being coordinates on a circle, tangent will take on all real values, showcasing its unique role across trigonometric concepts.