Problem 36 - Algebra 2

Question

Sketch each angle. Then find its reference angle. \(\frac{5 \pi}{4}\)

Step-by-Step Solution

Answer

The reference angle for \(\frac{5\pi}{4}\) is \(\frac{\pi}{4}\).

Step 1: Understand the Concept of Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. It is always positive and lies between 0 and \(\frac{\pi}{2}\). It helps in simplifying trigonometric calculations as it is the acute angle formed between the terminal side of an angle and the horizontal axis.

Step 2: Identify the Quadrant for \(\frac{5\pi}{4}\)

Determine which quadrant the angle \(\frac{5\pi}{4}\) lies in. Since it is between \(\pi\) (which is \(\frac{4\pi}{4}\)) and \(\frac{3\pi}{2}\) (which is \(\frac{6\pi}{4}\)), it lies in the third quadrant.

Step 3: Sketch the Angle \(\frac{5\pi}{4}\)

Begin from the positive x-axis and move clockwise or counter-clockwise depending on the given angle. For \(\frac{5\pi}{4}\), start from the positive x-axis and move 45 degrees (\(\frac{\pi}{4}\)) past \(\pi\). This will place the terminal side in the third quadrant.

Step 4: Compute the Reference Angle

Since \(\frac{5\pi}{4}\) is in the third quadrant, compute the reference angle by subtracting \(\pi\) from \(\frac{5\pi}{4}\). Therefore, the reference angle is \(\frac{5\pi}{4} - \pi = \frac{\pi}{4}\).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential in connecting angles to ratios of sides in right triangles. These functions—sine, cosine, tangent, and others—describe relationships between angle measures and distances, which become extremely useful in both geometry and analytical mathematics.

• **Sine and Cosine**: These functions are based on the unit circle (more on that in a bit). The sine of an angle is the y-coordinate, while the cosine is the x-coordinate of the point where the angle's terminal side intersects the unit circle.
• **Tangent**: This function can be found by taking the ratio of sine and cosine (i.e., tangent = sine/cosine). It represents the slope of a line depicted by the angle in standard position.

The beauty of trigonometric functions lies in their ability to be applied universally in various scenarios, irrespective of the size differences of triangles. Understanding these functions lets you draw connections between angles in diverse coordinate settings, like the unit circle, and facilitates deeper comprehension of spatial relationships in mathematics.

Navigating the Quadrants

In trigonometry, the coordinate plane is divided into four quadrants, each defined by the sign of the x and y coordinates:

• **First Quadrant**: Both x and y coordinates are positive. Here, all trigonometric functions are positive.
• **Second Quadrant**: The x coordinate is negative, and the y coordinate is positive. Sine is positive in this quadrant.
• **Third Quadrant**: Both x and y are negative. Tangent is the only function that is positive.
• **Fourth Quadrant**: The x coordinate is positive, and the y coordinate is negative. Cosine is positive here.

Knowing the quadrant where your angle resides can help determine the sign of the trigonometric functions for that angle. For example, in the case of the angle \(\frac{5\pi}{4}\), it lies in the third quadrant, meaning that sine and cosine will be negative, while tangent will remain positive. This is important because the reference angle found in this exercise, \(\frac{\pi}{4}\), would have different function values based on the quadrant.

Exploring the Unit Circle

The unit circle serves as a fundamental tool for understanding angles and trigonometric functions within a controlled system. It's a circle with a radius of one unit, centered at the origin of a coordinate plane. In this representation:

• The angle in standard position is measured from the positive x-axis counterclockwise.
• Each point on the unit circle corresponds to the cosine and sine of angles as mentioned before, making it invaluable for recalling or calculating sine and cosine values.

For \(\frac{5\pi}{4}\), placing this angle on the unit circle helps visualize its position, which is 45 degrees beyond \(\pi\), or 180 degrees (the negative x-axis), leading into the third quadrant. When sketching angles or solving problems, considering the unit circle helps simplify complex trigonometric scenarios, as it provides a straightforward visual reference that reveals the sign and values of trigonometric functions.