Problem 6 - Algebra and Trigonometry

Question

Find the reference angle for the given angle. $$ \begin{array}{llll}{\text { (a) } 99^{\circ}} & {\text { (b) }-199^{\circ}} & {\text { (c) } 359^{\circ}}\end{array} $$

Step-by-Step Solution

Answer

(a) 81°; (b) 19°; (c) 1°.

Step 1: Understand Reference Angle

A reference angle is the smallest positive angle that a given angle makes with the x-axis. It can be found in standard position for any given angle.

Step 2: Calculate Reference Angle for 99°

The angle 99° is in the second quadrant. To find the reference angle, subtract the angle from 180°: \[ \text{Reference Angle} = 180° - 99° = 81° \]

Step 3: Calculate Reference Angle for -199°

First, find the corresponding positive angle by adding 360° to -199°: \[ 361° - 199° = 161° \]Now, determine the reference angle in the second quadrant:\[ \text{Reference Angle} = 180° - 161° = 19° \]

Step 4: Calculate Reference Angle for 359°

The angle 359° is in the fourth quadrant. To find the reference angle, subtract it from 360°:\[ \text{Reference Angle} = 360° - 359° = 1° \]

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals primarily with the relationships and properties of triangles, especially right triangles. It focuses on the functions of angles, which are crucial not only in math but in various applications like physics, engineering, and even music theory. At the heart of trigonometry are the trigonometric functions such as sine, cosine, and tangent, which help in understanding the angle-related aspects of triangles.

• Sine, Cosine, Tangent: These basic trigonometric functions relate the angles and ratios of sides in a right triangle. For example, sine is the ratio of the opposite side to the hypotenuse, while cosine is the adjacent over the hypotenuse.
• Trigonometric Identities: Equations like Pythagorean identity, sine and cosine rules that hold true for all angles and are used to solve various trigonometric equations.
• Angles and Reference Angles: Understanding angles, especially how they are measured either in degrees or radians, is key. A reference angle, being the smallest angle to the x-axis, simplifies the use of sine, cosine, and tangent across different quadrants.

When learning trigonometry, it's essential to grasp how these functions and concepts interrelate, enabling more complex problem-solving scenarios. Breaking down angles into reference angles is particularly useful for simplification, as it leverages these foundational principles.

Quadrants

In trigonometry, the coordinate plane is divided into four regions called quadrants. These quadrants help in determining the signs of trigonometric functions for different angles. The quadrants are numbered counterclockwise starting from the positive x-axis. Each quadrant has specific characteristics and understanding them is crucial for navigating angles correctly.

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, while cosine and tangent are negative. The reference angle for angles here is calculated by subtracting the angle from 180°.
• Third Quadrant (180° to 270°): Tangent is positive, whereas sine and cosine are negative. Adjusting angles here involves subtracting 180° from the angle.
• Fourth Quadrant (270° to 360°): Cosine is positive, and sine and tangent are negative. Reference angles are found by subtracting the angle from 360°.

Understanding these quadrants not only helps in determining reference angles but also in figuring out the signs of trigonometric functions at these angles. This knowledge is particularly helpful in simplifying and solving trigonometric equations.

Standard Position

An angle is said to be in standard position if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. This setup is fundamental in trigonometry as it helps provide uniformity to measuring and analyzing angles. By understanding the standard position, one can more easily interpret angle positions and their trigonometric functions.

• Initial Side and Terminal Side: The angle starts from the initial side (along the x-axis) and opens to the terminal side, determining its measure.
• Positive and Negative Angles: Positive angles are measured counterclockwise from the initial side, while negative angles are measured clockwise.
• Angle Measurement: Measured in degrees or radians, where 360° equals 2π radians.

Having angles in the standard position simplifies visualizing where they fall on the coordinate plane, which subsequently aids in identifying their quadrants and calculating reference angles. Therefore, mastering the concept of standard position allows for more effective analysis of angles and enhances problem-solving skills in trigonometry.