Problem 7 - Algebra 2

Question

The given point \(P\) is located on the unit circle. Find \(\sin \theta\) and \(\cos \theta\) $$ P\left(-\frac{3}{5}, \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

\( \cos \theta = -\frac{3}{5} \), \( \sin \theta = \frac{4}{5} \).

Step 1: Identify the Coordinates

The point given is \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \). These coordinates represent the \( x \) and \( y \) coordinates on the unit circle. On the unit circle, the formula for a point is \( (\cos \theta, \sin \theta) \).

Step 2: Determine \( \cos \theta \)

From the coordinates of point \( P \), we see that the \( x \)-coordinate is \( -\frac{3}{5} \). Therefore, \( \cos \theta = -\frac{3}{5} \).

Step 3: Determine \( \sin \theta \)

Similarly, the \( y \)-coordinate of point \( P \) is \( \frac{4}{5} \). Therefore, \( \sin \theta = \frac{4}{5} \).

Step 4: Verify the Point is on the Unit Circle

Verify by checking if the equation \( x^2 + y^2 = 1 \) holds true for \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \). Calculate: \( \left(-\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 = \frac{9}{25} + \frac{16}{25} = 1 \). Since this equals 1, the point is confirmed to be on the unit circle.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of trigonometry, which is the branch of mathematics that studies the properties of triangles and the relationship between their sides and angles.

These functions are intimately connected to the unit circle and include the sine (\( \sin \theta \)) and cosine (\( \cos \theta \)) functions. These two are the most basic trigonometric functions and relate to the coordinates on the unit circle.

• Sine Function (\( \sin \theta \)): In a right triangle, it represents the ratio of the length of the opposite side to the hypotenuse. On the unit circle, \( \sin \theta \) corresponds to the \( y \)-coordinate of a point.
• Cosine Function (\( \cos \theta \)): Represents the ratio of the length of the adjacent side to the hypotenuse. On the unit circle, \( \cos \theta \) is the \( x \)-coordinate of a point.

Each angle in the circle is represented by a point in \( (x, y) \) format, where \( x = \cos \theta \) and \( y = \sin \theta \). By understanding sine and cosine through the unit circle, you see how they can describe an angle using coordinates.

Coordinates on the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Its unique property is that every point on the unit circle represents the angle formed with the positive x-axis.

In the context of trigonometric functions, the coordinates of a point on the unit circle \( (x, y) \) directly illustrate the following:

• The \( x \)-coordinate is the cosine of the angle \( \theta \) (i.e., \( \cos \theta \)).
• The \( y \)-coordinate corresponds to the sine of the angle \( \theta \) (i.e., \( \sin \theta \)).

For example, in the problem provided, the point \( P \) is \( \left(-\frac{3}{5}, \frac{4}{5}\right) \). This means that \( \cos \theta = -\frac{3}{5} \) and \( \sin \theta = \frac{4}{5} \).

By translating angles to positions on the unit circle, we can solve complex problems involving trigonometry in a visual and geometric manner.

Pythagorean Identity

The Pythagorean Identity is a critical mathematical relationship in trigonometry, derived from the Pythagorean Theorem. It connects the cosine and sine of an angle in a simple equation:\[ \cos^2 \theta + \sin^2 \theta = 1 \]This equation holds true for every angle \( \theta \), at any point on the unit circle.

It’s a powerful tool as it adds a layer of verification when working with trigonometric functions. Applying this identity helps verify that the point does in fact sit on the unit circle. For instance, with point \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \), you calculate:

• \( \left(-\frac{3}{5}\right)^2 = \frac{9}{25} \)
• \( \left(\frac{4}{5}\right)^2 = \frac{16}{25} \)
• Adding these, \( \frac{9}{25} + \frac{16}{25} = 1 \)

Since the sum is 1, this confirms that the coordinates describe a point on the unit circle, reinforcing the derived trigonometric values for \( \sin \theta \) and \( \cos \theta \). The Pythagorean Identity thus plays a key role in understanding and utilizing the unit circle effectively in trigonometry.