Problem 30 - Algebra 2 and Trigonometry

Question

Angle \(R O P\) is an angle in standard position with \(\mathrm{m} \angle R O P=\theta, R(1,0)\) a point on the initial side of \(\angle R O P\) and \(P(p, q)\) the point at which the terminal side of \(\angle R O P\) intersects the unit circle. a. What is the domain of \(\cos \theta\) and of \(\sin \theta ?\) b. What is the range of \(\cos \theta\) and of \(\sin \theta\) ? c. Is tan \(\theta\) defined for all angle measures? d. What is the domain of \(\tan \theta ?\) e. What is the range of \(\tan \theta ?\)

Step-by-Step Solution

Answer

a. Domain of both \(\cos \theta\) and \(\sin \theta\) is all real numbers. b. Range of both \(\cos \theta\) and \(\sin \theta\) is \([-1, 1]\). c. No, \(\tan \theta\) is not defined for all angles. d. Domain of \(\tan \theta\) is all reals except \(\theta = \frac{\pi}{2} + k\pi\). e. Range of \(\tan \theta\) is all real numbers.

Step 1: Understand the Unit Circle

The unit circle is a circle with radius 1 centered at the origin (0, 0) in the coordinate plane. Any angle in standard position can have its terminal side intersect the unit circle at the point \((p, q)\). Here, \(p = \cos \theta\) and \(q = \sin \theta\).

Step 2: Determine the Domain of cosine and sine

Since angle \(\theta\) in standard position can rotate an infinite number of times around the circle, \(\theta\) can be any real number. Therefore, both \(\cos \theta\) and \(\sin \theta\) have the domain of all real numbers.

Step 3: Determine the Range of cosine and sine

On the unit circle, \(\cos \theta\) represents the x-coordinate and \(\sin \theta\) represents the y-coordinate of the point \((p,q)\). They range from -1 to 1 on the unit circle. Thus, the range for both \(\cos \theta\) and \(\sin \theta\) is \([-1, 1]\).

Step 4: Determine if Tangent is defined for all angles

The tangent of an angle \(\theta\) is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Tangent is undefined when \(\cos \theta = 0\), such as at \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. Therefore, \(\tan \theta\) is not defined for these angle measures.

Step 5: Determine the Domain of tangent

Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) becomes undefined when \(\cos \theta = 0\), the domain excludes these points. Thus, the domain of \(\tan \theta\) is all real numbers except \(\theta = \frac{\pi}{2} + k\pi\).

Step 6: Determine the Range of tangent

The values of \(\sin \theta\) and \(\cos \theta\) can cause \(\tan \theta\) (defined as \(\frac{\sin \theta}{\cos \theta}\)) to approach both positive and negative infinity. Hence, the range of \(\tan \theta\) is all real numbers \((-\infty, \infty)\).

Key Concepts

Unit Circle

The unit circle is fundamental in understanding trigonometric functions. It is a circle with a radius of exactly one, centered at the origin (0, 0) on a coordinate plane. This geometric tool is essential because it helps illustrate how the sine and cosine functions work. You can imagine any angle in standard position, starting from the positive x-axis, having a terminal side that sweeps out and eventually intersects the unit circle.

• The x-coordinate of this intersection point corresponds to \( \cos \theta \).
• The y-coordinate corresponds to \( \sin \theta \).

Using the unit circle, it becomes clear why the values of sine and cosine are constrained to between -1 and 1, as these coordinates describe positions on the unit circle. Moreover, it showcases how trigonometric concepts are tied to geometrical visualizations, making them easier to understand and apply in various mathematical problems.

Domain and Range

For the cosine and sine functions, it is essential to grasp the idea of domains and ranges, which are foundational when dealing with functions. The domain refers to the set of all possible input values. In trigonometric terms, since an angle \( \theta \) in standard position can rotate freely, the domain of both \( \cos \theta \) and \( \sin \theta \) is the set of all real numbers.
In contrast, the range focuses on the output that these functions can yield. In the unit circle, the cosine function takes on all x-values while sine takes on all y-values from -1 to 1. Thus, the range for \( \cos \theta \) and \( \sin \theta \) is \([-1, 1]\).
These terms help set the boundaries within which these trigonometric functions operate, establishing their limits based on geometrical constraints.

Sine and Cosine Functions

The sine and cosine functions are two of the most crucial tools in trigonometry. These periodic functions help describe oscillations and waves over a continuous spectrum. Each can be directly derived from the unit circle:

• \( \cos \theta \) gives the x-coordinate of a point on the unit circle, which varies according to the angle \( \theta \).
• \( \sin \theta \) provides the y-coordinate of the same point.

Both sine and cosine functions are periodic with a period of \( 2\pi \), meaning they repeat their values every \( 2\pi \) radians. Understanding these cycles can assist in predicting and calculating behavior for various applications, such as sound waves, alternating current electricity, and mechanical vibrations.
Fully appreciating sine and cosine functions allows for better engagement with cyclical patterns seen in nature and technology.

Tangent Function

The tangent function, another important trigonometric function, differs from sine and cosine since it combines both. Defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), it describes the slope of the line formed by the terminal side of an angle in standard position.

• Tangent is not always defined, specifically when \( \cos \theta = 0 \), such as at angles like \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
• These points cause the function to have vertical asymptotes, infinite lines that the tangent graph approaches but never touches.

Thus, the domain is all real numbers except where these zeroes occur, creating discontinuities. However, the range of the tangent function is unrestricted, spanning from minus infinity to plus infinity \( (-\infty, \infty) \).
Understanding the tangent function is vital in mathematical concepts involving rates of change, slopes, and angles, providing another layer of insight into dynamic systems.