Problem 16 - Precalculus: Functions and Graphs

Question

Let \(P\) be the point on the unit circle \(U\) that corresponds to \(t .\) Find the coordinates of \(P\) and the exact values of the trigonometric functions of \(t,\) whenever possible. (a) \(7 \pi / 4\) (b) \(-3 \pi / 4\)

Step-by-Step Solution

Answer

(a) \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\), (b) \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Any angle \(t\) measured in radians corresponds to a point \((x, y)\) on the unit circle, where \(x = \cos(t)\) and \(y = \sin(t)\).

Step 2: Locate the Angle for Part (a)

The angle \(t = \frac{7\pi}{4}\) is equivalent to \(2\pi - \frac{\pi}{4}\). This means the terminal side of the angle is in the fourth quadrant, where cosine is positive and sine is negative.

Step 3: Find Coordinates and Trigonometric Functions for Part (a)

On the unit circle, the reference angle is \(\frac{\pi}{4}\), for which \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). In the fourth quadrant, the coordinates of \(P\) are \((\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\). Thus, \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

Step 4: Locate the Angle for Part (b)

The angle \(t = -\frac{3\pi}{4}\) means rotating clockwise from the positive x-axis. This places the terminal side of the angle in the third quadrant, where both cosine and sine are negative.

Step 5: Find Coordinates and Trigonometric Functions for Part (b)

The reference angle for \(-\frac{3\pi}{4}\) is \(\frac{3\pi}{4}\), which corresponds to a third quadrant angle. Here, \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\). Thus the coordinates of \(P\) are \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\) and \(\cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \(\sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, they describe the relationship between an angle, measured in radians, and the coordinates of a point on the unit circle.
In the unit circle:

• Cosine (\(\cos(t)\)) is the x-coordinate of the point on the unit circle corresponding to an angle \(t\).
• Sine (\(\sin(t)\)) is the y-coordinate of that same point.
• Tangent (\(\tan(t)\)) is the ratio of sine to cosine, or \(\frac{\sin(t)}{\cos(t)}\).

These functions are fundamental in mathematics, particularly in geometry and trigonometry, as they extend to describe the oscillations and waves in physics, and solve problems involving right-angled triangles. Understanding how these functions map angles to coordinates on the unit circle is essential for grasping trigonometric principles.

Radian Measure

Radian measure is a way to express angles by describing the distance traveled along a circle's circumference relative to its radius. Unlike degrees, which are divided into 360 parts, radians divide the circle into \(2\pi\) parts.

• One complete revolution around the circle represents an angle of \(2\pi\) radians.
• \(\pi\) radians corresponds to 180 degrees.
• Half a circle, or \(\pi\) radians, is equivalent to a semicircle.

The key advantage of radians is that they simplify many mathematical formulas. For understanding rotations and cycles, like those represented on the unit circle, radians provide a natural and meaningful way to describe angles.

Quadrants on the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the positive or negative sign of the coordinates. Understanding these quadrants helps us know the behavior of trigonometric functions.
The quadrants are divided as follows:

• First Quadrant: Both x and y coordinates are positive (\(0\) to \(\frac{\pi}{2}\) radians).
• Second Quadrant: x is negative, y is positive (\(\frac{\pi}{2}\) to \(\pi\) radians).
• Third Quadrant: Both x and y coordinates are negative (\(\pi\) to \(\frac{3\pi}{2}\) radians).
• Fourth Quadrant: x is positive, y is negative (\(\frac{3\pi}{2}\) to \(2\pi\) radians).

Each quadrant provides unique characteristics for cosine and sine based on the signs of x and y. Recognizing which quadrant an angle's terminal side lies in helps quickly determine the signs of the trigonometric functions corresponding to that angle.