Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \tan \frac{5 \pi}{4} $$

Step-by-Step Solution

Answer

\( \tan \frac{5\pi}{4} = 1 \)

Step 1: Understand the Angle

The angle given is \( \frac{5\pi}{4} \). We need to find where this angle lies on the unit circle. \( \frac{5\pi}{4} \) radians is in the third quadrant of the unit circle since it falls between \( \pi \) and \( \frac{3\pi}{2} \).

Step 2: Recall the Unit Circle Properties

In the unit circle, tangent is defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). In the third quadrant, both sine and cosine are negative, thus \( \tan(\theta) \) will be positive.

Step 3: Use the Unit Circle

\( \tan(\theta) \) for angles that are odd multiples of \( \frac{\pi}{4} \) is \(1\) in absolute value terms. Since \( \frac{5\pi}{4} = \pi + \frac{\pi}{4} \), this is an angle of \( \frac{\pi}{4} \) advanced into the third quadrant, retaining a tangent value of \(1\) but positive due to same nature of sine and cosine.

Step 4: Conclusion on Tangent Value

Since the tangent function in the third quadrant for angles like \( \frac{5\pi}{4} \) results in \( \tan \theta = 1 \), the exact value of \( \tan \frac{5\pi}{4} \) is \(1\).

Key Concepts

Unit Circle

The unit circle is a vital concept in trigonometry and helps in understanding how trigonometric functions work. It is a circle with a radius of 1, centered at the origin of a coordinate system (0,0). On the unit circle, every angle in standard position corresponds to a point, which has coordinates

• x = \( \cos(\theta) \)
• y = \( \sin(\theta) \)

The angle \( \theta \) is measured from the positive x-axis counterclockwise. This allows us to define the sine and cosine of an angle precisely, using the coordinates on the unit circle.
Moreover, understanding the quadrants is essential:

• In Quadrant I: Both sine and cosine are positive.
• In Quadrant II: Sine is positive, cosine is negative.
• In Quadrant III: Both sine and cosine are negative.
• In Quadrant IV: Sine is negative, cosine is positive.

This knowledge aids in understanding how the tangent function behaves in different quadrants, particularly relevant for solving problems like finding \( \tan \frac{5\pi}{4} \).

Tangent Function

The tangent function, often written as \( \tan(\theta) \), is one of the fundamental trigonometric functions. It can be expressed in terms of the sine and cosine functions: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \] The value of \( \tan(\theta) \) depends on the values of sine and cosine for a given angle \( \theta \).

• In Quadrant I: \( \tan(\theta) \) is positive, since both \( \sin(\theta) \) and \( \cos(\theta) \) are positive.
• In Quadrant II: \( \tan(\theta) \) is negative, as \( \sin(\theta) \) is positive but \( \cos(\theta) \) is negative.
• In Quadrant III: \( \tan(\theta) \) becomes positive again because both \( \sin(\theta) \) and \( \cos(\theta) \) are negative, making their ratio positive.
• In Quadrant IV: \( \tan(\theta) \) is negative again, with a negative \( \sin(\theta) \) and a positive \( \cos(\theta) \).

Understanding this behavior especially helps with angles like \( \frac{5\pi}{4} \), which falls in the third quadrant and thus has a positive tangent value.

Radians

Radians are an alternative to degrees for measuring angles and are crucial in many fields of mathematics. A radian measures how far along the circumference of a circle the angle takes you, where one full circle equals \( 2\pi \) radians. In practical terms:

• \( \pi \) radians is half a circle (180 degrees)
• \( \frac{\pi}{2} \) radians is a quarter circle (90 degrees)
• \( \pi \) is equivalent to 180 degrees
• \( 2\pi \) is a full circle, equating to 360 degrees

Using radians simplifies many calculations in trigonometry, such as converting angles and calculating trigonometric functions.
When we consider \( \frac{5\pi}{4} \), we recognize that it is slightly larger than \( \pi \) and less than \( \frac{3\pi}{2} \), placing it comfortably in the third quadrant of the unit circle.
Working with radians can initially be confusing if you're used to degrees, but it's a more natural unit when it comes to circles and trigonometry.