Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Find the reference angle \(\theta_{\mathrm{k}}\) if \(\theta\) has the given measure. (a) \(3 \pi / 4\) (b) \(4 \pi / 3\) (c) \(-\pi / 6\) (d) \(9 \pi / 4\)

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{4} \), (b) \( \frac{\pi}{3} \), (c) \( \frac{\pi}{6} \), (d) \( \frac{\pi}{4} \).

Step 1: Define Reference Angle

A reference angle is the positive acute angle that a given angle forms with the x-axis. It is always measured in the positive direction, and its value is between 0 and \( rac{ ext{π}}{2} \).

Step 2: Determine Quadrant for \( \theta = \frac{3\pi}{4} \)

The angle \( \frac{3\pi}{4} \) falls in the second quadrant because it is less than \( \pi \) but greater than \( \frac{\pi}{2} \). In the second quadrant, the reference angle is \( \pi - \theta \).

Step 3: Calculate Reference Angle for \( \theta = \frac{3\pi}{4} \)

Subtract \( \theta \) from \( \pi \) to get the reference angle: \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).

Step 4: Determine Quadrant for \( \theta = \frac{4\pi}{3} \)

The angle \( \frac{4\pi}{3} \) falls in the third quadrant since it is greater than \( \pi \) but less than \( \frac{3\pi}{2} \). In the third quadrant, the reference angle is \( \theta - \pi \).

Step 5: Calculate Reference Angle for \( \theta = \frac{4\pi}{3} \)

Subtract \( \pi \) from \( \theta \) to find the reference angle: \( \frac{4\pi}{3} - \pi = \frac{\pi}{3} \).

Step 6: Determine Quadrant for \( \theta = -\frac{\pi}{6} \)

Since the angle \( -\frac{\pi}{6} \) is negative, we find the positive equivalent by adding \( 2\pi \), resulting in \( 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \), which is in the fourth quadrant. In this quadrant, the reference angle is \( 2\pi - \theta \).

Step 7: Calculate Reference Angle for \( \theta = -\frac{\pi}{6} \)

First, find the positive equivalent angle, \( \frac{11\pi}{6} \). Then calculate the reference angle: \( 2\pi - \frac{11\pi}{6} = \frac{\pi}{6} \).

Step 8: Simplify \( \theta = \frac{9\pi}{4} \) to a Coterminal Angle

Since \( \frac{9\pi}{4} \) is greater than \( 2\pi \), subtract \( 2\pi \) (or \( \frac{8\pi}{4} \)) to find a coterminal angle: \( \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4} \).

Step 9: Determine Reference Angle for \( \theta = \frac{9\pi}{4} \)

Using the coterminal angle \( \frac{\pi}{4} \), which is already between 0 and \( \frac{\pi}{2} \), the reference angle is \( \frac{\pi}{4} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and can be incredibly helpful when working with angles. Essentially, it is a circle with a radius of one, centered at the origin of a coordinate system. This circle is used to define the trigonometric functions for all angles.
Why is it a crucial part of understanding trigonometry? Because it allows you to easily visualize and compute the values of sine, cosine, and tangent for different angles. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, while the y-coordinate corresponds to the sine of the angle.
The unit circle is also broken down into four quadrants. Each quadrant covers 90 degrees or \( \frac{\pi}{2} \) radians. Knowing the location of an angle on the unit circle helps determine the sign and value of the trigonometric functions quickly.

Radians

Radians are a way to measure angles. Unlike degrees, which are divided into 360 parts for a full circle, radians relate the angle to the radius and circumference of the circle.
A full circle is \(2\pi\) radians. Therefore:

• \(\pi\) radians is half a circle, or 180 degrees.
• \(\frac{\pi}{2}\) radians is a quarter of a circle, or 90 degrees.
• 1 radian is about 57.3 degrees.

In trigonometry, angles are often expressed in radians because this unit is grounded in the mathematical properties of the circle, making it more convenient for calculations on the unit circle.
When solving exercises with angles expressed in radians, it is important to be comfortable converting between radians and degrees, especially when determining the reference angle or its equivalent in different quadrants.

Trigonometric Functions

Trigonometric functions are functions that relate angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, and tangent. These functions are pivotal in understanding the behavior of angles on the unit circle.
- **Sine (\( \sin \))**: It represents the y-coordinate of a point on the unit circle.- **Cosine (\( \cos \))**: It represents the x-coordinate of a point on the unit circle.- **Tangent (\( \tan \))**: It is the ratio of sine to cosine, or the slope of the line that intersects the point on the unit circle.Understanding these functions is key to determining the values attributed to different angles and solving problems related to them. These functions and their reciprocals can help to find unknown angles or sides in triangles, analyze periodic patterns, and much more.

Quadrants

The coordinate plane on which the unit circle is placed is divided into four quadrants. Each quadrant influences the sign of the coordinates, and thus the sign of trigonometric functions.
- **First Quadrant**: Both x and y are positive. Therefore, sine, cosine, and tangent are all positive.- **Second Quadrant**: x is negative, y is positive. Here, sine is positive while cosine and tangent are negative.- **Third Quadrant**: Both x and y are negative. Tangent is positive (since the negatives cancel out), while sine and cosine are negative.- **Fourth Quadrant**: x is positive, y is negative. Cosine is positive, but sine and tangent are negative.
Knowing in which quadrant an angle lies is crucial because it affects the computation of the reference angle and the signs of its trigonometric values. This is particularly important when evaluating the reference angle of standard angles such as \(3\pi/4\) or \(-\pi/6\), as these angles determine the necessary adjustments needed to compute accurate values.