Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ \left(-\frac{\sqrt{5}}{3},-\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

a. \(\sin \theta = -\frac{2}{3}\), b. \(\cos \theta = -\frac{\sqrt{5}}{3}\), c. \(\tan \theta = \frac{2\sqrt{5}}{5}\), d. \(\sec \theta = -\frac{3\sqrt{5}}{5}\), e. \(\csc \theta = -\frac{3}{2}\), f. \(\cot \theta = \frac{\sqrt{5}}{2}\).

Step 1: Understanding Unit Circle Coordinates

In the context of the unit circle, the coordinates of a point on the circle are given by \((x, y)\), where \(x = \cos \theta\) and \(y = \sin \theta\). Given the coordinates \((-\frac{\sqrt{5}}{3}, -\frac{2}{3})\), it follows that \(\cos \theta = -\frac{\sqrt{5}}{3}\) and \(\sin \theta = -\frac{2}{3}\).

Step 2: Finding \( \sin \theta \)

Using the information from Step 1, we find \(\sin \theta = -\frac{2}{3}\).

Step 3: Finding \( \cos \theta \)

From the coordinates, we already have \(\cos \theta = -\frac{\sqrt{5}}{3}\).

Step 4: Calculating \( \tan \theta \)

The tangent of the angle \(\theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\). Thus, \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = \frac{2}{\sqrt{5}}\). This can be rationalized to \(\frac{2\sqrt{5}}{5}\).

Step 5: Finding \( \sec \theta \)

The secant of \(\theta\) is the reciprocal of \(\cos \theta\). Hence, \(\sec \theta = -\frac{3}{\sqrt{5}} = -\frac{3\sqrt{5}}{5}\) (after rationalization).

Step 6: Finding \( \csc \theta \)

The cosecant of \(\theta\) is the reciprocal of \(\sin \theta\). Thus, \(\csc \theta = -\frac{3}{2}\).

Step 7: Finding \( \cot \theta \)

The cotangent of \(\theta\) is the reciprocal of \(\tan \theta\). From \(\tan \theta = \frac{2\sqrt{5}}{5}\), the cotangent is \(\frac{\sqrt{5}}{2}\).

Step 8: Final Answers

Using the steps above, here are the values:- \(\sin \theta = -\frac{2}{3}\)- \(\cos \theta = -\frac{\sqrt{5}}{3}\)- \(\tan \theta = \frac{2\sqrt{5}}{5}\)- \(\sec \theta = -\frac{3\sqrt{5}}{5}\)- \(\csc \theta = -\frac{3}{2}\)- \(\cot \theta = \frac{\sqrt{5}}{2}\)

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of geometry, especially in dealing with right triangles. Some of the core trigonometric functions include:

• Sine ( \( \sin \theta \)) - which is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine ( \( \cos \theta \)) - which is the ratio of the adjacent side to the hypotenuse.
• Tangent ( \( \tan \theta \)) - defined as the ratio of sine to cosine, or the opposite side to the adjacent side.

In context of the unit circle, these functions correspond directly to the coordinates of a point on the circle. Here, they help in understanding angles and distances:

• The x-coordinate of a point on the unit circle is \( \cos \theta \).
• The y-coordinate is \( \sin \theta \).
• The tangent can then be represented as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Every trigonometric function can be associated with the position of a point on the unit circle, which is a critical aspect of how these functions are applied beyond triangles.

Standard Position Angle

An angle is said to be in standard position if its vertex is at the origin of the coordinate plane and its initial side (the side from which the angle starts) lies along the positive x-axis. This is a common way to position angles when dealing with the unit circle, as it provides a consistent method to measure angles.
Here are a few important points about standard position angles:

• Measured counterclockwise, angles are positive; measured clockwise, they are negative.
• The terminal side is the other side of the angle, which rotates from the initial side.
• The measure of the angle, \( \theta \), determines where the terminal side lies relative to the x-y plane.

When a point like \((-\frac{\sqrt{5}}{3}, -\frac{2}{3})\) is mentioned, this means that the terminal side intersects the unit circle at point \(P\) with coordinates equivalent to \(\cos \theta\) and \(\sin \theta\). Understanding this concept is essential for solving problems that involve moving within the circle and calculating distances, angles, or other measurements.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are defined as follows:

• Cosecant ( \( \csc \theta \)) - the reciprocal of sine, given by \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant ( \( \sec \theta \)) - the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent ( \( \cot \theta \)) - reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \) or \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These functions are especially useful when dealing with trigonometric identities and solving problems involving trigonometric equations. For instance:

• If \( \sin \theta = -\frac{2}{3} \), then \( \csc \theta = -\frac{3}{2} \).
• Similarly, with \( \cos \theta = -\frac{\sqrt{5}}{3} \), \( \sec \theta = -\frac{3\sqrt{5}}{5} \).
• For \( \tan \theta = \frac{2\sqrt{5}}{5} \), \( \cot \theta = \frac{\sqrt{5}}{2} \).

Grasping these reciprocal relationships helps in deepening understanding of how the functions interrelate and provide solutions using inverse measures.

Coordinate Geometry

Coordinate geometry is a branch of geometry where the position of points on the plane is described using an ordered pair of numbers. In the context of trigonometry, coordinate geometry is crucial for visualizing angles and functions on the unit circle.
The unit circle is a perfect example of coordinate geometry in action. It is a circle centered at the origin (0,0) with radius of one unit. Here’s how it connects to trigonometric functions:

• Coordinates \((x, y)\) on the circle represent the cosine and sine of a given angle taken from the positive x-axis.
• Thus, any point on the unit circle can be interpreted in terms of \( \cos \theta \) and \( \sin \theta \).
• For example, for the point \((-\frac{\sqrt{5}}{3}, -\frac{2}{3})\), \(x = \cos \theta \) and \(y = \sin \theta \).

Understanding these connections enables solving complex trigonometric problems effectively, as it provides a visual representation and association of angles and functions, making abstract mathematical concepts more tangible and logical.