Problem 7 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Quadrant

Identify the quadrant of the angle \(-\frac{3 \pi}{4}\). This angle is negative, which means we're moving in the clockwise direction starting from 0. The magnitude is greater than \(\pi/2\) but less than \(\pi\), so we end up in the second quadrant where both sine and cosine are negative.

Step 2: Find Reference Angle

Find the reference angle by taking the absolute value of the given angle. The reference angle for \(-\frac{3 \pi}{4}\) is \(\frac{3 \pi}{4}\). In the unit circle, this angle corresponds to the point \(-\sqrt{2}/2, -\sqrt{2}/2\).

Step 3: Compute Tangent

Compute the tangent by dividing the y-coordinate by the x-coordinate. Therefore, \(\tan(-\frac{3 \pi}{4}) = -\sqrt{2}/2 / -\sqrt{2}/2 = 1\).

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions, represented as \( \tan \theta \). It is defined as the ratio of the sine and cosine of the same angle \( \theta \). For any angle \( \theta \), this can be expressed mathematically as:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This function is periodic with a period of \( \pi \), meaning it repeats every \( \pi \) radians. It is essential to note that the tangent function is undefined when the cosine of \( \theta \) is zero since division by zero is undefined.

• In intervals of \( \pi/2 \), such as \( \pi/2 \), \( 3\pi/2 \), etc., the function remains undefined because the x-value on the unit circle is zero.
• Tangent values can be positive or negative depending on the quadrant in which \( \theta \) is located.
• The graph of the tangent function is characterized by its recurring vertical asymptotes at intervals of \( \pi \).

Understanding the properties of the tangent function is crucial in solving various trigonometric problems, including those involving angles on the unit circle.

Unit Circle

The unit circle is a powerful tool in trigonometry that helps us visualize the relationships between angles and their trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.

• The circle encompasses all angles measured in radians, with 0 located at the positive x-axis.
• As we move counterclockwise around the circle, we increase the angle, while moving clockwise decreases it.
• Any point on the unit circle can be identified by its coordinates, \((\cos \theta, \sin \theta)\).

These coordinates are fundamental because they allow us to determine the sine and cosine of any angle \( \theta \). The unit circle not only aids in evaluating trigonometric functions but also helps determine the sign of these functions based on the angle’s position in a specific quadrant.

Reference Angle

A reference angle is the acute angle formed by the terminal side of an angle \( \theta \) and the horizontal axis. It is always a positive angle measured from 0 to \( \pi/2 \) radians (or 0 to 90 degrees).

• Reference angles are useful for simplifying the calculations of trigonometric functions, as they allow us to find equivalent acute angles.
• To find the reference angle for a given angle \( \theta \), one typically takes the absolute value of \( \theta \) and adjusts to ensure the angle is acute.

For instance, with \(-\frac{3 \pi}{4}\), the reference angle is \( \frac{3 \pi}{4}\), which effectively points us to the same trigonometric values apart from the sign, allowing us to determine functions like tangent more easily.

Trigonometric Quadrants

The unit circle is divided into four quadrants, each of which has unique properties that affect the sign of trigonometric functions.

• First Quadrant (0 to \( \pi/2 \)): All trigonometric functions are positive.
• Second Quadrant (\( \pi/2 \) to \( \pi \)): Sine is positive, while cosine and tangent are negative.
• Third Quadrant (\( \pi \) to \( 3\pi/2 \)): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (\( 3\pi/2 \) to \( 2\pi \)): Cosine is positive, while sine and tangent are negative.

Understanding these quadrant properties is important when solving problems involving the tangent function, as they determine the sign of the result. In the example of \(-\frac{3 \pi}{4}\), the angle resides in the second quadrant, indicating why the tangent of this angle is positive, as both coordinates are negative and their division yields a positive value.