Problem 47 - Algebra 2

Question

Show that cos \(A\) defined as a ratio equals cos \(\theta\) using the unit circle.

Step-by-Step Solution

Answer

The cosine of an angle \(A\) defined as a ratio equals the cosine of an angle \(\theta\) on a unit circle because in both cases the cosine is determined by the same geometry and ratio: the ratio of the length of the side adjacent to the angle to the hypotenuse in the case of the ratio definition, which equals the x-coordinate of the point on the circle in the case of the unit circle definition. The angle used does not change the result.

Step 1: Recall the Definition of Cosine

Cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, \(\cos(A) = \frac{adjacent}{hypotenuse}\).

Step 2: Refer to the Unit Circle

A unit circle is a circle with radius 1. In a unit circle, an angle \(\theta\) in standard position has its vertex at the origin, its initial side on the positive x-axis, and its terminal side passing through a point (x, y) on the circle. The cosine of \(\theta\) is defined as the x-coordinate of this point, i.e., \(\cos(\theta) = x\). This is because in a unit circle, the length of the hypotenuse (which is the radius) is 1. So, the ratio of the adjacent side to the hypotenuse will be the same as the length of the adjacent side itself (as the ratio to 1 doesn't change the value).

Step 3: Show the Equality

Given that the definitions of \(\cos(A)\) and \(\cos(\theta)\) are both reliant on the same geometry, and the ratio in the first definition is preserved in the second definition due to the radius of the unit circle being 1, we can therefore say that \(\cos(A) = \cos(\theta) = x\). The angle used does not change the result, so the cosines of \(A\) and \(\theta\) are equivalent when using the ratio definition and the unit circle definition.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry, focusing on the ratio of the sides of a right triangle.

• The cosine of an angle, for instance, angle \( A \), is the ratio of the adjacent side to the hypotenuse of the triangle.
• This fundamental trigonometric function helps to measure how much of the angle's "shadow" falls along the adjacent side compared to the hypotenuse.

From a mathematical perspective, the cosine function is often expressed as \( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
In various practical applications, this function helps determine distances or angles in diverse fields such as physics, engineering, and even computer graphics. It is important to understand that cosine, as part of a right triangle, helps determine precise geometric properties, linking it inherently to the unit circle.

Unit Circle

The unit circle is a central concept that simplifies understanding trigonometric functions.

• It is a circle with a radius of 1, centered at the origin of a coordinate plane.
• All points on the unit circle satisfy the equation \( x^2 + y^2 = 1 \).

This unique property allows the entire circle to express any angle measured from the positive x-axis.
The unit circle redefines the intuitive nature of the cosine function by associating the x-coordinate of a point on the circle's circumference with \( \cos(\theta) \).
Given the circle has a radius of 1, the hypotenuse of any right triangle formed is always unity; hence, \( \cos(\theta) \) is simply the x-coordinate of the respective terminal arm of the angle \( \theta \). By leveraging this property, one can evaluate cosine values for angles not traditionally found in right triangles.

Angle Measurement

In trigonometry, angles can be measured in various units, with degrees and radians being the most common.

• Degrees divide a circle into 360 parts, where each part represents one degree.
• Radians offer a more natural measurement related to the circle's geometry, where one full circle equals \( 2\pi \) radians.

These measurements provide different perspectives on angle size and utility depending on the context.
When referencing angles in the unit circle, radians are typically favored due to their simplicity, especially in higher mathematics.
The equivalency between degrees and radians is expressible through conversion, such as \( 180^\circ = \pi \) radians.
Understanding angle measurement is vital when analyzing rotational motion or describing periodic phenomena, where both cosine and the unit circle play critical roles. In summary, mastering angle measurement ensures precise calculations of trigonometric functions like cosine, as they rely heavily on the representation of angles in both radian and degree forms.