Problem 52 - Precalculus: A Concise Course

Question

Find the reference angle \(\boldsymbol{\theta}^{\prime}\), and sketch \(\boldsymbol{\theta}\) and \(\boldsymbol{\theta}^{\prime}\) in standard position. $$ \theta=11.6 $$

Step-by-Step Solution

Answer

The reference angle θ' is 11.6 degrees, the same as the given angle θ. They are represented by the same sketch, as they both fall in the first quadrant.

Step 1: Identify the Quadrant in which θ lies

The given angle θ = 11.6 degrees is a positive acute angle. It lies between 0 and 90 degrees, which means it is found in the first quadrant.

Step 2: Determine the Value of Reference Angle (θ')

In the case of the first quadrant, the reference angle (θ') equals the angle itself. Therefore, the reference angle θ' is 11.6 degrees.

Step 3: Sketch the Given Angle θ and the Reference Angle θ'

To sketch the angle θ, start from zero degree line (positive x-axis), then rotate counterclockwise 11.6 degrees. Considering this, the line indicating the angle will lie in the first quadrant. Since θ' equals θ in the first quadrant, both sketches will be the same.

Key Concepts

Understanding Trigonometry

Trigonometry, a branch of mathematics, is crucial in understanding the relationships between the angles and sides of triangles. It is deeply connected to geometric principles and often comes into play when dealing with circles and cyclical motions. At its core, trigonometry focuses on six primary functions: sine (sin), cosine (cos), tangent (tan), along with their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).

These trigonometrical functions are vital for solving problems involving right-angle triangles, where one angle is precisely 90 degrees. With any of the non-right angles (acute angles) given, trigonometry allows us to determine any unknown side or angle within the triangle. Furthermore, these functions have been extended to all angles, whether positive or negative and irrespective of their magnitude, by defining them as ratios in a unit circle framework.Trigonometry also has numerous real-world applications, from architecture and engineering to computer graphics and astronomy. A foundational understanding of angles and their interaction with trigonometric functions is a prerequisite for grasping the full depth of trigonometry.

Standard Position of an Angle

When discussing the standard position of an angle in trigonometry, we envision the initial side of the angle lying along the positive x-axis, and the vertex situated at the origin (0,0) of the coordinate plane. Rotation from the initial side gives us the terminal side.

An angle is considered to be in a standard position when these criteria are met. Positive angles are drawn with counterclockwise rotation, while negative angles involve a clockwise rotation. Moreover, identifying the standard position is crucial to understand reference angles, which is the acute angle that the terminal side of an angle in standard position makes with the x-axis. It is always a non-negative angle, no greater than 90 degrees.Understanding the standard position becomes even more significant when projecting trigonometric functions onto the coordinate plane, leading to the concept of the unit circle, which in turn helps in visualizing the trigonometric functions' values based on the terminal side's location.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Starting from the positive x-axis and rotating counterclockwise, the quadrants are numbered from I to IV:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Each quadrant holds significance in trigonometry, as the signs of the trigonometrical functions change depending on which quadrant an angle's terminal side lies in. For example, in Quadrant I, all the trigonometric functions yield positive values, but in Quadrant II, only sine and cosecant are positive. Understanding how angles are situated in these quadrants and how trigonometric functions adapt accordingly is essential for solving problems and fully understanding trigonometric relations.