Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(\theta\) be an angle in standard position with \((x, y)\) a point on the terminal side of \(\theta\) and \(r=\sqrt{x^{2}+y^{2}} \neq 0 .\) Fill in the blank. \(\sin \theta=\) ____.

Step-by-Step Solution

Answer

\(\sin \theta = \frac{y}{r}\)

Step 1: Understanding what is \(\sin \theta\) in the Unit Circle

In the unit circle, the sine of an angle \(\theta\) is defined as the y-coordinate of the point where the terminal side of the angle \(\theta\) intersects the unit circle. That is, \(\sin \theta = y\).

Step 2: Expressing \(\sin \theta\) in terms of x, y and r

However, we may need to relate \(\sin \theta\) to the radius r. From a given \(r=\sqrt{x^{2}+y^{2}} \neq 0\), we can express y in terms of r and x. But in the case of \(\sin \theta\), it is directly defined as the ratio of y to r from the right triangle formed in the unit circle. Thus, we can express \(\sin \theta\) as the ratio of y to r, i.e., \(\sin \theta = \frac{y}{r}\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that deals with angles and triangles. It helps in finding relationships between the angles and sides of a right triangle. In trigonometry, the sine of an angle \( \theta \) is the ratio of the length of the opposite side to the hypotenuse in a right triangle. \
This function is commonly used in various fields, such as physics, engineering, and astronomy. To find the sine of an angle \( \theta \), you can use:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Within the context of the unit circle, the sine of an angle is specifically defined as the y-coordinate where the terminal side of the angle intersects the circle. So, when given a point \( (x, y) \) on the unit circle, \( \sin \theta \) is equal to the y-coordinate, which helps in understanding the position of the angle.

Unit Circle

The unit circle is a vital concept in trigonometry, representing a circle located in the coordinate plane with a radius of one unit. This circle simplifies calculations and offers visual insights into trigonometric functions. It is centered at the origin, which is the point (0,0). Because the radius is 1, the parameter for any point (x, y) on this circle at an angle \( \theta \) from the positive x-axis can be found as follows:

• The x-coordinate is \( \cos \theta \)
• The y-coordinate is \( \sin \theta \)

These coordinates reveal how the sine and cosine functions correspond to any given angle. By visualizing the unit circle, one can easily comprehend the behavior of sine and cosine as periodic functions, which repeat every 360 degrees or 2\(\pi\) radians. Moreover, it provides a straightforward way to observe how angles larger than 90 degrees relate to their corresponding coordinates on the circle.

Angle in Standard Position

An angle in standard position offers a consistent and standardized method to describe an angle's measure. It ensures that the analysis of angles is uniform and easily comparable across problems.
An angle is said to be in standard position when:

• Its vertex is located at the origin of the coordinate system.
• Its initial side lies along the positive x-axis.
• It sweeps out towards the positive direction, usually counterclockwise.

This concept is extremely useful because it allows the angle to interact with both the unit circle and trigonometric functions in a cohesive and predictable manner. It's typically measured in degrees or radians, with a full rotation around the circle amounting to 360 degrees or 2\( \pi \) radians. Understanding angles in this framework helps in the interpretation and calculation of sine, cosine, and other trigonometric function values.