Problem 16 - Algebra and Trigonometry

Question

In Exercises \(5-18,\) the unit circle has been divided into twelve equal arcs, corresponding to t-values of $$ 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{5 \pi}{6}, \pi, \frac{7 \pi}{6}, \frac{4 \pi}{3}, \frac{3 \pi}{2}, \frac{5 \pi}{3}, \frac{11 \pi}{6}, \text { and } 2 \pi $$ Use the \((x, y)\) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined. $$ \cos \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The value of \(\cos \frac{3 \pi}{2}\) is 0

Step 1: Understand the unit circle

The unit circle is a circle with a radius of 1, centered at the origin (\(0, 0\)) of a coordinate plane. It is commonly used in trigonometry, and the \(x\)-coordinate of a point on the unit circle gives the cosine of the angle defined by the line segment from the origin to the point, and the positive x-axis.

Step 2: Locate the position on the unit circle

\(\frac{3 \pi}{2}\) corresponds to the point on the unit circle where the line from the origin to the point makes an angle of \(\frac{3 \pi}{2}\) with the positive x-axis. This is equivalent to 270 degrees in the unit circle, which corresponds to the point on the negative y-axis.

Step 3: Apply definition of cosine

Cosine of an angle in the unit circle is defined as the \(x\)-coordinate of the point on the unit circle. For the position at \(\frac{3 \pi}{2}\) or 270 degrees, the \(x\)-coordinate is 0.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential in understanding relationships within a right-angled triangle and can also be extended to functions on a unit circle. In particular, the main functions are sine, cosine, and tangent, which relate angles to ratios of two side lengths.
However, on the unit circle, we use these functions to represent these relationships in terms of coordinates rather than lengths.

• **Sine (sin)** represents the y-coordinate on the unit circle.
• **Cosine (cos)** is associated with the x-coordinate on the unit circle.
• **Tangent (tan)** is the ratio of the sine and cosine, computed as \( \frac{\sin}{\cos} \).

Understanding these fundamentals helps in efficiently working with angles and measurements on the coordinate plane. Each function provides a unique way to interpret and calculate positional relationships on the circle.

Exploring the Cosine Function

The cosine function on a unit circle is crucial because it provides the x-coordinate value for any given angle. In essence, as you move around the circle, the x-coordinates of the points where you land define the cosine values for those angles.
To picture this, imagine a line from the center of the circle moving outward; where it intersects the circle, the x-coordinate represents the cosine.

• For an angle \( \theta \), cosine is calculated as the adjacent side over hypotenuse in a right triangle context. But on the unit circle, it's simply the x-coordinate of the intersecting point.
• **At \( \frac{3\pi}{2} \)**, which is where the line extends straight down from the origin vertically, the x-coordinate will be 0.

This understanding of how cosine corresponds to the x-values on the unit circle helps unravel helpful properties about periodic functions.

Navigating the Coordinate Plane

The coordinate plane, consisting of the x-axis and y-axis, provides a framework to analyze geometric and algebraic concepts. Dividing the plane into four quadrants, each quadrant corresponds to specific combinations of positive and negative x and y values.
The unit circle is a perfect specimen of the plane, with its center at the origin \( (0, 0) \), and an emphasis on the relationship between angles and coordinates, specifically:

• The x-coordinate represents horizontal distances on the plane.
• The y-coordinate measures vertical distances.

When observing points on the circle, the x and y coordinates indicate specific trigonometric values, offering an insightful way to explore angles beyond the traditional limits of a right-angle triangle.

Approaching Angle Measurement

Angle measurement on a unit circle is typically denoted in radians, a natural and intuitive way of expressing angles, especially in higher mathematics. For practical use, knowing common angles in radians is beneficial, such as \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \text{and } 2\pi\).

• One full circle is \(2\pi\) radians, correlating to 360 degrees.
• A half circle (straight line) is \(\pi\) radians or 180 degrees.
• The right angle is \(\frac{\pi}{2}\) radians, equivalent to 90 degrees.

Understanding these basic fractions helps in converting between radians and degrees and provides a fundamental understanding of angle navigation on a coordinate plane. So for \(\frac{3\pi}{2}\), this angle corresponds to 270 degrees, aligning directly with the negative y-axis on the unit circle.