Problem 5 - Algebra and Trigonometry

Question

Find the reference angle for the given angle. $$ \begin{array}{llll}{\text { (a) } 225^{\circ}} & {\text { (b) } 810^{\circ}} & {\text { (c) }-105^{\circ}}\end{array} $$

Step-by-Step Solution

Answer

(a) 45°, (b) 90°, (c) 75°.

Step 1: Understanding Reference Angles

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. Reference angles are always positive and between \(0^{\circ}\) and \(90^{\circ}\), inclusive.

Step 2: Find Reference Angle for 225°

The angle 225° is in the third quadrant. To find its reference angle, subtract 180°: \[ 225^{\circ} - 180^{\circ} = 45^{\circ} \]. Thus, the reference angle is 45°.

Step 3: Normalize 810° for Full Revolutions

To find the reference angle for 810°, first bring it within a full circle (0° to 360°) by finding the remainder when dividing by 360°: \[ 810^{\circ} \mod 360^{\circ} = 90^{\circ} \].

Step 4: Find Reference Angle for the Reduced 90°

The angle 90° is on the positive y-axis, and when an angle is on the axis, it serves as its own reference angle. Thus, the reference angle is 90°.

Step 5: Normalize -105° for Standard Range

Since -105° is negative, add 360° to find its equivalent positive angle: \[ -105^{\circ} + 360^{\circ} = 255^{\circ} \].

Step 6: Find Reference Angle for 255°

The angle 255° is in the third quadrant. To find its reference angle, subtract 180° from it: \[ 255^{\circ} - 180^{\circ} = 75^{\circ} \]. Thus, the reference angle is 75°.

Key Concepts

Angle Measurement

Understanding angle measurement is fundamental in trigonometry and geometry. Angles are typically measured in degrees or radians, with degrees being more common in basic trigonometry.
Measuring angles helps us determine the position of a line relative to another, usually the horizontal x-axis. Here, a full circle is divided into 360 degrees. When we refer to an angle like 225°, we're describing an angle that has been measured counterclockwise from the positive x-axis.

• If an angle exceeds 360°, it means the angle completes at least one full rotation.
• Negative angles are measured clockwise from the positive x-axis.

This cyclic nature of angles is important as it helps in converting angles beyond the customary ranges through addition or subtraction of full circles (360°). For example, an angle of 810° can be reduced to its equivalent, 90°, by recognizing that it surpasses the 360° mark enough times to leave a remainder of 90°.

Trigonometric Angles

A trigonometric angle refers to any angle measured in standard position, meaning it has its vertex at the origin and its initial side along the positive x-axis. These angles are incredibly useful in solving various mathematical and real-world problems, including those involving periodic functions like sine and cosine.
Reference angles play a critical role in this context. They are always positive and between 0° and 90°, serving as a bridge to find the values of trigonometric functions for angles that lie in different quadrants. For instance, knowing that 225° is in the third quadrant, we can find the reference angle using the formula:

• Reference Angle (RA) = Angle - 180° (for angles in the third quadrant).

Hence, for 225°, the reference angle comes out to be 45°. This reference angle can then be utilized to calculate the sine, cosine, and tangent because these functions inherently reflect according to the quadrant in which the original angle lies.

Quadrants in Trigonometry

The coordinate plane is divided into four quadrants in trigonometry, which help to determine the sign of trigonometric functions based on the location of the angle's terminal side.
From a starting position along the positive x-axis, angles measured can fall into any of these four sections:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, while cosine and tangent are negative.
• Third Quadrant (180° to 270°): Only tangent is positive; sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, while sine and tangent are negative.

Recognizing the quadrants becomes essential, especially when dealing with angles like -105°, which initially seems complex. By normalizing -105° by adding 360°, we get 255°, placing it firmly in the third quadrant. Knowing that only tangent remains positive here helps in understanding how the trigonometric functions will behave for this angle.