Problem 10 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$\frac{11 \pi}{4}$$

Step-by-Step Solution

Answer

The reference angle for the given angle \(11 \pi / 4\) is \(\pi / 4\).

Step 1: Remove the extra full rotations

Given that the angle is \(11 \pi / 4\). Notice this angle has completed more than one full circle of \(2 \pi\) (or \(8 \pi / 4\)). By subtracting the integer multiples of \(2 \pi\) we get the angle in the interval \(0 < θ < 2 \pi\). Here, we subtract \(2 \pi \) (divide \(2 \pi\) into 4 parts to have common denominator) two times from the given angle (since \(2 \pi < 11 \pi / 4 < 3 \pi\)) which give us \((11 \pi / 4) - 2*(2 \pi) = 11 \pi / 4 - 8 \pi / 4 = 3 \pi /4\). So, \(3 \pi /4\) is the associated angle within the first rotation of the given angle.

Step 2: Find the reference angle

The reference angle is always the acute angle with respect to the x-axis. Here, \(3 \pi / 4\) lies in the second quadrant. The reference angle for second quadrant is given by the formula \(θ' = θ - π/2\). But since the calculated angle is more than \(π/2\) it need to be subtracted from \(\pi\). So, reference angle will be \(\pi - θ = \pi - 3 \pi / 4 = \pi / 4\).

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and lengths of triangles, particularly right-angled triangles. The fundamental functions of trigonometry are sine, cosine, and tangent, which are the ratios of the sides of a right triangle. These functions are essential in various fields, including physics, engineering, and even music theory.

When solving problems related to angles, such as finding the reference angle, it's crucial to understand these trigonometric functions and how they relate to the angles on the unit circle, which can be measured in degrees or radians. Understanding how to determine the reference angle is particularly useful in simplifying calculations and making it easier to evaluate trigonometric functions. The reference angle corresponds to its acute angle made with the x-axis, and it allows for the use of standard values of trigonometric functions to solve problems.

The Unit Circle and its Importance

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. Its importance in trigonometry cannot be overstated, as it provides a visual and conceptual way to understand the behavior of trigonometric functions over different angles. The circumference of the unit circle is tied to the concept of radians and is equal to a distance of \(2\pi\) units, representing a full rotation of 360 degrees.

The unit circle is divided into four quadrants, and each point on the circle's circumference corresponds to an angle formed by the radius with the positive x-axis. The coordinates of these points (\(cos\theta, sin\theta\)) are the values of the cosine and sine functions for any given angle \(\theta\). Reference angles are used in conjunction with the unit circle because they provide a way to relate obtuse or reflex angles back to acute angles, for which the values of trigonometric functions are commonly known and easier to work with.

Radians: A Measure of Angles

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. One radian is the angle created when the radius of the circle is wrapped along the circumference, and a full circle is \(2\pi\) radians, equivalent to 360 degrees. The radian measure is often preferred in mathematics because it provides a direct relationship between the length of an arc of a circle and the angle that subtends that arc.

To convert from degrees to radians, we multiply by \(\frac{\pi}{180}\), and to convert back, we multiply by \(\frac{180}{\pi}\). In the context of the provided exercise, we are working with radians, and it becomes evident when we need to subtract full rotations of \(2\pi\) from an initial angle to find its associated angle within the first rotation. This helps to determine the reference angle, which is helpful when working with trigonometric functions as it simplifies calculations by using positive acute angles, which are easier to handle.