Problem 43 - Algebra 2

Question

a. Geometry Draw a unit circle on the coordinate plane. Then draw five angles in standard position measuring \(\frac{\pi}{5}, \frac{4 \pi}{5}, \frac{6 \pi}{5}, \frac{9 \pi}{5},\) and \(\frac{3 \pi}{10}\) radians. b. For each angle, complete a right triangle. Place the hypotenuse along the terminal side (from the origin to the unit circle). Place one leg along the \(x\) -axis. The other leg will be parallel to the \(y\) -axis. c. Critical Thinking. Are the five triangles congruent? Justify your answer by using the values of \(\sin \theta\) and \(\cos \theta\) for each angle.

Step-by-Step Solution

Answer

The triangles will not all be congruent, as the \(\sin \theta\) and \(\cos \theta\) values differ for the different angles given. These values depend on the specific theta value in the unit circle, which varies for each angle, thus resulting in different dimensions for the right triangles. Hence, it can be inferred that the triangles are not congruent.

Step 1: Drawing a Unit Circle and Angles

Draw a unit circle on a coordinate plane. Mark the five angles in standard position, which are \(\frac{\pi}{5}, \frac{4 \pi}{5}, \frac{6 \pi}{5}, \frac{9 \pi}{5},\) and \(\frac{3 \pi}{10}\). Standard position means that the initial side of the angle is on the positive x-axis, and the angle rotates counterclockwise.

Step 2: Drawing Right Triangles

For each of the identified angles, complete a right triangle. The hypotenuse of the triangle should be placed along the terminal side (the line extending from the origin to the point on the unit circle), one leg along the x-axis, and the other leg parallel to the y-axis. This way, we have formed five right triangles.

Step 3: Analysis of Triangle Congruency

To ascertain whether the five triangles formed are congruent, we must analyze the \(\sin \theta\) and \(\cos \theta\) for each angle. The \(\sin \theta\) value of an angle in a right triangle is the length of the side opposite the angle (y-coordinate) divided by the hypotenuse (always 1 in a unit circle), while the \(\cos \theta\) value is the length of the adjacent side (x-coordinate) divided by the hypotenuse. By calculating the sine and cosine values for each angle, we can deduce whether the triangles are congruent. Congruent triangles will have the same sine and cosine values.

Key Concepts

Radians

Radians are a way to measure angles based on the radius of a circle. Instead of using degrees, radians provide a more natural and mathematical approach. One full rotation around a circle is equal to \(2\pi\) radians, which is approximately 6.28318.

This unit ties directly to the concept of a circle since the circumference of a circle is \(2\pi r\), where \(r\) is the radius. Thus, in a unit circle, the circumference is simply \(2\pi\), making calculations straightforward.

Angles measured in radians often appear in forms like \(\frac{\pi}{5}\) or \(\frac{3\pi}{10}\). These expressions show a fraction of the circle's total circumference.

• This makes it easier to understand rotations and arc lengths without extra conversions.
• Most trigonometric calculations are simpler in radians.

Understanding radians means recognizing how they relate to the circle and how effortlessly they connect trigonometric concepts.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees, or \(\frac{\pi}{2}\) radians. This special configuration allows us to explore various trigonometric relationships.

When you draw right triangles on a unit circle, we utilize the x-axis and y-axis to form the triangle's legs, with the hypotenuse lying along the circle's radius.

• The hypotenuse is always 1 in a unit circle, simplifying the calculations.
• The right triangle helps us visualize key trigonometric functions.

These triangles have direct implications on understanding sine and cosine as they directly correspond to the y and x coordinates, respectively. Once you grasp the layout of right triangles on the circle, identifying these functions becomes much easier.

This diagrammatic approach helps deeply connect angles to their side lengths.

Trigonometric Functions

Trigonometric functions like sine and cosine arise from examining the properties of right triangles within the unit circle.

In a right triangle drawn in the unit circle:

• The sine function, \(\sin \theta\), relates to the length of the side opposite the angle \(\theta\).
• The cosine function, \(\cos \theta\), relates to the length of the adjacent side.

The hypotenuse is always 1 in a unit circle, making \(\sin \theta\) the y-coordinate and \(\cos \theta\) the x-coordinate for the terminal point of the angle.

Understanding these functions is crucial as they provide the foundation for more complex trigonometric relationships and applications.

Trigonometric functions extend beyond simple angles and assist in modeling everything from waves to oscillations in both science and engineering.

Their deep integration into mathematics shows their significance and utility across various fields.

Congruent Triangles

Triangles are congruent if they have the same size and shape. In terms of right triangles on a unit circle, we check for congruency using their sine and cosine values.

Since the hypotenuse is always 1, congruency focuses on whether the opposite and adjacent sides are identical.

• If \(\sin \theta\) and \(\cos \theta\) are the same for two triangles, they are congruent.
• Different sine or cosine values indicate differences in their angles or side ratios.

This exercise reveals that it's more than visual symmetry; mathematical values determine congruency. While different angles can lead to congruent triangles, altering just one aspect can break congruency.

By understanding congruency, one can ensure precise constructions and calculations, leading to reliable outcomes in problem-solving.