Problem 14 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(95^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(85^{\circ}\).

Step 1: Determine the Quadrant

First, observe that the given angle is \(95^{\circ}\). Since it is between \(90^{\circ}\) and \(180^{\circ}\), it lies in the second quadrant.

Step 2: Calculate the Reference Angle

In the second quadrant, the reference angle \( \theta_r \) is found by subtracting the angle from \(180^{\circ}\). Thus, \( \theta_r = 180^{\circ} - 95^{\circ} = 85^{\circ} \).

Step 3: Expression of the Reference Angle

Write your reference angle result for consistency: \( \theta_r = 85^{\circ} \).

Key Concepts

Quadrant Classification

In trigonometry, understanding quadrant classification is crucial when dealing with angles. When an angle is drawn in standard position, you determine which quadrant it lies in by its measure. The coordinate plane is divided into four quadrants:

• First Quadrant: Angles here measure from 0° to 90°.
• Second Quadrant: Angles measure from 90° to 180°.
• Third Quadrant: Angles extend from 180° to 270°.
• Fourth Quadrant: Angles range from 270° to 360°.

Knowing which quadrant an angle lies in helps predict certain properties of the angle, such as its sine, cosine, and tangent values. In our example, the angle is 95°. Because 95° falls between 90° and 180°, we determine it is situated in the second quadrant.

Angle Measurement

Measuring angles is a fundamental skill in trigonometry. Angles can be measured in degrees or radians, but here we're focusing on degrees since the exercise uses this unit. A full circle is 360°, which is divided into four quadrants as explained earlier. An angle in standard position always starts from the positive x-axis and moves counter-clockwise. For example, an angle of 95° starts from the positive x-axis and then extends 95° in the counter-clockwise direction. When you move through these degrees, it's crucial to understand how your position changes across each quadrant, as this affects both trigonometric ratios and the calculation of reference angles. Understanding angle measurement ensures you can accurately determine both the angle's quadrant and its reference angle.

Second Quadrant

The second quadrant is particularly interesting because of its specific characteristics in trigonometry and geometry. Whenever an angle is in this quadrant, its sine function is positive while the cosine and tangent functions are negative. This is due to the position on the coordinate plane where y-values are positive and x-values are negative.To find the reference angle of an angle in the second quadrant, you need to subtract the angle from 180°. This is important because the reference angle is the acute angle the terminal side forms with the x-axis. Using the exercise example, where the angle is 95°:- The reference angle is calculated by subtracting from 180°: \[ \theta_r = 180^{\circ} - 95^{\circ} = 85^{\circ} \]This 85° reference angle is used to simplify the understanding of trigonometric functions in this standard position, making it much easier to work with angles in different quadrants. Recognizing the quadrant and calculating the reference angle allows one to determine the signs and values of trigonometric functions efficiently.