Problem 3 - Algebra 2

Question

Use the graph of \(y=\tan \theta\) to find each value. If the tangent is undefined at that point, write undefined. $$ \tan \frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

The value of \( tan \frac{3 \pi}{4} \) is -1.

Step 1: Understanding Tan function

The function \( y = tan \theta \) is a periodic function with a period of \( \pi \), which means it repeats its pattern every \( \pi \) units. It's undefined for \( \theta = \pi(0.5k) \) where \( k \) is an odd integer. The function is positive in the first and third quadrants, and negative in the second and fourth quadrants.

Step 2: Locate the angle on the unit circle

The given angle \( \theta \) is \( \frac{3 \pi}{4} \), which falls in the second quadrant. The special angles in the second quadrant are \( \frac{\pi}{2} \) and \( \frac{2 \pi}{3} \), and as our given angle is \( \frac{3 \pi}{4} \), it is between these two values.

Step 3: Find the tangent

Since the angle is in the second quadrant, where tan function values are negative, and it is closer to \( \frac{\pi}{2} \) than \( \frac{2 \pi}{3} \), we can anticipate that the value of \( tan \frac{3 \pi}{4} \) would be negative but close to 0, since the value of the tangent goes from \( + \infty \) to 0 when \( \theta \) goes from \( 0 \) to \( \pi/2 \), and from 0 to \( - \infty \) when \( \theta \) goes from \( \pi/2 \) to \( \pi \). In other words, the sign of the tangent depends on the quadrant in which the angle \( \theta \) falls, and as our given angle is in the second quadrant, \( tan \theta \) is negative. However, the exact value of \( tan \frac{3 \pi}{4} \) is -1.

Key Concepts

Understanding the Unit Circle

A fundamental concept in trigonometry is the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a crucial tool for understanding and applying trigonometric functions.

• The unit circle allows us to easily understand the values of sine, cosine, and tangent at different angles.
• Angles on the unit circle are often measured in radians, where the entire circle represents an angle of \(2\pi\) radians.
• Each point on the unit circle corresponds to a cosine and sine of that angle, represented as \((\cos \theta, \sin \theta)\).

When using the unit circle to find tangent values, remember that the tangent of an angle \(\theta\) is the ratio of the sine to the cosine at that angle: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This understanding helps us determine whether the tangent is defined or undefined at various points on the unit circle.

Tangent Function Characteristics

The tangent function, \(y = \tan \theta\), is one of the primary trigonometric functions.

• It is periodic, repeating its values every \(\pi\) radians.
• The tangent function is undefined where \(\cos \theta = 0\) because it involves division by zero, specifically at \(\theta = \frac{\pi}{2} + k\pi\) for any integer \(k\).

Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), its values depend heavily on the behavior of \(\sin \theta\) and \(\cos \theta\). Unlike sine and cosine, the tangent can take any real value and does not have a bounded range. This makes it unique among trigonometric functions and particularly useful in different applications. For angles where the tangent is defined, its sign and value also depend on the angle's position on the unit circle.

The Role of Quadrants

Understanding which quadrant an angle falls into is crucial when working with trigonometric functions like tangent.

• The unit circle is divided into four quadrants, each corresponding to specific angle ranges:
  • First Quadrant: \(0\) to \(\frac{\pi}{2}\)
  • Second Quadrant: \(\frac{\pi}{2}\) to \(\pi\)
  • Third Quadrant: \(\pi\) to \(\frac{3\pi}{2}\)
  • Fourth Quadrant: \(\frac{3\pi}{2}\) to \(2\pi\)
• The sign of the tangent function is determined by the quadrant:
• Positive in the first and third quadrants.
• Negative in the second and fourth quadrants.

For the given exercise, the angle \(\frac{3\pi}{4}\) is in the second quadrant. Here, the tangent function values are negative. Knowing this, you can predict the sign of the tangent even before doing the exact calculation. Thus, the value of \(\tan \frac{3\pi}{4}\) is negative, and more specifically, it is \(-1\). This insight simplifies the understanding of how trigonometric values behave across the different regions of the unit circle.