Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Find the reference angle \(\theta_{\mathrm{k}}\) if \(\theta\) has the given measure. (a) \(7 \pi / 4\) (b) \(2 \pi / 3\) (c) \(-3 \pi / 4\) (d) \(-23 \pi / 6\)

Step-by-Step Solution

Answer

(a) \(\pi / 4\), (b) \(\pi / 3\), (c) \(\pi / 4\), (d) \(\pi / 6\)

Step 1: Understanding Reference Angles

A reference angle is the smallest angle that can be made from the terminal side of the given angle and the x-axis. Its value is always between 0 and \(\pi\) radians (or 0 and 180 degrees).

Step 2: Finding Reference Angle for \(7 \pi / 4\)

The angle \(7 \pi / 4\) is in the fourth quadrant because it is more than \(3\pi / 2\) but less than \(2\pi\). The reference angle \(\theta_{\mathrm{k}}\) is \(2\pi - 7\pi / 4 = 8\pi / 4 - 7\pi / 4 = \pi / 4\).

Step 3: Finding Reference Angle for \(2 \pi / 3\)

The angle \(2\pi / 3\) is in the second quadrant because it is more than \(\pi / 2\) but less than \(\pi\). The reference angle \(\theta_{\mathrm{k}}\) is \(\pi - 2\pi / 3 = 3\pi / 3 - 2\pi / 3 = \pi / 3\).

Step 4: Finding Reference Angle for \(-3 \pi / 4\)

The angle \(-3\pi / 4\) is equivalent to \(\pi - 3\pi / 4 = 4\pi / 4 - 3\pi / 4\) in the positive direction, making it in the second quadrant. The reference angle \(\theta_{\mathrm{k}}\) is again \( \pi / 4 \).

Step 5: Finding Reference Angle for \(-23 \pi / 6\)

First, convert \(-23\pi / 6\) into a positive angle. It is equal to \(23\pi / 6 - 4\cdot\pi = 23\pi / 6 - 24\pi / 6 = -\pi / 6\). Thus, in positive terms (as \(-\pi / 6\) from \(2\pi\) is the same place as \(2\pi - \pi / 6\)), the reference angle is \(\pi / 6\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between angles and sides in triangles. It's particularly focused on right-angled triangles, but its principles extend beyond just triangles and into the realm of circular functions.

By understanding trigonometry:

• You can solve practical problems involving triangles and circles.
• You will learn about sine, cosine, and tangent functions, which help in defining angle relationships.
• It uses radians and degrees as units for angles, aiding calculations in various fields like physics and engineering.

Grasping these foundational concepts can greatly aid in understanding more complex mathematical and scientific computations, especially when dealing with wave functions and oscillations.

Quadrants

In a coordinate plane, the space is divided into four sections, known as quadrants. Each quadrant houses specific ranges of coordinates, impacting the evaluation of trigonometric functions.

The quadrants are:

• First Quadrant: Here, both x and y coordinates are positive ( (+, +) ).
• Second Quadrant: The x coordinate is negative, and the y is positive ( (-, +) ).
• Third Quadrant: Both coordinates are negative ( (-, -) ).
• Fourth Quadrant: X is positive, and y is negative ( (+, -) ).

Trigonometric functions behave differently in each quadrant due to the sign changes. Understanding quadrants is crucial, as it helps to determine angles' reference angles and assess sine and cosine functions.

Radian measure

Radian measure offers a standardized unit for measuring angles, bringing a high level of precision. Unlike degrees, radians relate directly to the arc length and radius of a circle, making it a pivotal unit in mathematics.

Consider this:

• 1 radian is the angle formed when the radius is wrapped around the circle's edge.
• The full circle encompasses 2π radians, equivalent to 360 degrees.
• Often used in calculus due to its simplicity in derivative calculations.

Using radians can simplify mathematical expressions and is a compact way to express angles, especially in the context of trigonometric functions and equations.

Reference angle calculation

The reference angle is a critical concept that simplifies trigonometric calculations by reducing any given angle to an acute angle. It is the smallest angle a terminal angle makes with the x-axis. The reference angle between 0 and π/2 simplifies the complexity of trigonometric equations.

Here’s how you can find a reference angle:

• If the angle is in the first quadrant: The reference angle is the angle itself.
• Second quadrant: (π - ext{angle})
• Third quadrant: ( ext{angle} - π)
• Fourth quadrant: (2π - ext{angle})

Employing reference angles allows for simplified computation, especially when dealing with angles greater than 2π or negative angles. By converting these angles to their reference equivalents, you maintain accuracy while also easing the calculation process.