Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-16: 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) \(5 \pi / 4\) (b) \(-\pi / 4\)

Step-by-Step Solution

Answer

(a) \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\), \(\sin=\cos=-\frac{\sqrt{2}}{2}\), \(\tan=1\); (b) \((\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\), \(\sin=-\frac{\sqrt{2}}{2}\), \(\cos=\frac{\sqrt{2}}{2}\), \(\tan=-1\).

Step 1: Understanding the Unit Circle

The unit circle is a circle in the coordinate plane with a radius of 1 and centered at the origin (0,0). Each point on the unit circle corresponds to an angle \( t \), measured in radians, from the positive x-axis.

Step 2: Determine Coordinates for 5π/4

For the angle \( t = \frac{5\pi}{4} \), since \( \frac{5\pi}{4} \) is in the third quadrant, the point \( P \) on the unit circle will have both x and y coordinates negative. The reference angle is \( \frac{\pi}{4} \), so the coordinates are \( (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) \).

Step 3: Determine Trigonometric Functions for 5π/4

The sine and cosine functions for \( \frac{5\pi}{4} \) are equal to the y and x coordinates respectively. Therefore, \( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \) and \( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \). Tangent is given by the ratio \( \tan(\frac{5\pi}{4}) = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \).

Step 4: Determine Coordinates for -π/4

For \( t = -\frac{\pi}{4} \), this corresponds to an angle \( \frac{7\pi}{4} \) when looked at in the positive direction. This angle lies in the fourth quadrant, meaning the x-coordinate is positive and the y-coordinate is negative. The coordinates are \( (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) \).

Step 5: Determine Trigonometric Functions for -π/4

The sine and cosine functions for \( -\frac{\pi}{4} \) are the y and x coordinates, respectively. Thus, \( \sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \) and \( \cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \). The tangent is \( \tan(-\frac{\pi}{4}) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding circular motion and oscillations. They relate the angles of a triangle within a circle to the ratios of the triangle's sides. In the unit circle, which has a radius of 1, these functions are defined as follows:

• Sine (\( ext{sin}\)) is the y-coordinate of a point on the unit circle.
• Cosine (\( ext{cos}\)) is the x-coordinate of that same point.
• Tangent (\( ext{tan}\)) is the ratio of the y-coordinate to the x-coordinate (\( ext{tan}(t) = \frac{\text{sin}(t)}{\text{cos}(t)}\)).

Understanding these functions helps to not only identify the position on the unit circle but also to evaluate other trigonometric identities.

Coordinates on Unit Circle

The unit circle is a perfect circle centered at the origin of a Cartesian plane with a radius of 1. This allows for easy computation of trigonometric functions as every point on the circumference represents an angle, measured in radians from the positive x-axis.

• The coordinates of any point \(P(t)\) can be found using \( ( ext{cos}(t), ext{sin}(t))\).
• For example, for an angle of \( \frac{5\pi}{4}\), the coordinates are \( (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\).
• When the angle is \(-\frac{\pi}{4}\), it translates to \( (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\) as it lies in the fourth quadrant.

Reference Angle

A reference angle is the smallest angle \( \theta \) formed with the x-axis by the terminal side of an angle \( t \) in standard position. It's always measured between 0 and \( \frac{\pi}{2}\) radians (or 0 and 90 degrees).

• For the angle \( \frac{5\pi}{4}\), the reference angle is \( \frac{\pi}{4} \).
• This reference angle helps to determine the sine and cosine values by using fundamental trigonometric identities for angles located in standard position.
• Reference angles simplify finding trigonometric values for angles located in different quadrants by providing a common angle base.

Understanding reference angles can simplify and unify the process of solving trigonometric problems.

Quadrants

The coordinate plane is divided into four sections, known as quadrants. Each of these quadrants is defined by the signs of the x and y coordinates:

• The first quadrant: both x and y coordinates are positive.
• The second quadrant: x is negative and y is positive.
• The third quadrant: both x and y coordinates are negative.
• The fourth quadrant: x is positive and y is negative.

Angles in trigonometry take different coordinate signs depending on which quadrant they terminate in. For example:

• \( \frac{5\pi}{4}\) lies in the third quadrant, leading to negative sine and cosine values,
• \(-\frac{\pi}{4}\) is situated in the fourth quadrant, resulting in a positive cosine but a negative sine.

Understanding which quadrant an angle lies in helps to accurately determine the signs of trigonometric functions.