Problem 23 - Algebra 2 and Trigonometry

Question

In \(18-27\) , express each given function value in terms of a function value of a positive acute angle (the reference angle). \(\sin 285^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 285^{\circ} = -\sin 75^{\circ}\).

Step 1: Identify the Reference Angle

First, identify in which quadrant the angle \(285^{\circ}\) lies. Since \(285^{\circ}\) is between \(270^{\circ}\) and \(360^{\circ}\), it is in the fourth quadrant. The reference angle \(\theta\) is found by subtracting \(285^{\circ}\) from \(360^{\circ}\): \(\theta = 360^{\circ} - 285^{\circ} = 75^{\circ}\).

Step 2: Understand Sine in Quadrants

In the fourth quadrant, the sine function is negative. This means that \(\sin 285^{\circ}\) is the negative of the sine of its reference angle.

Step 3: Express In Terms of Reference Angle

Use the reference angle found in Step 1. \(\sin 285^{\circ} = -\sin 75^{\circ}\).

Key Concepts

Reference Angle

Understanding the idea of a reference angle is essential when working with trigonometric functions. A reference angle is the smallest angle the terminal side of the given angle makes with the x-axis. It is always a positive acute angle, meaning it falls between 0 and 90 degrees.

To find the reference angle for any given angle, follow these guidelines:

• If the angle is in the first quadrant, the reference angle is the same as the given angle.
• In the second quadrant, subtract the angle from 180 degrees.
• For the third quadrant, subtract the angle from 180 degrees.
• In the fourth quadrant, subtract the angle from 360 degrees.

For our problem, \(285^\circ\) lies in the fourth quadrant. The reference angle is \(360^\circ - 285^\circ = 75^\circ\). This reference angle helps us relate a given angle to an acute angle.

Sine Function

The sine function is one of the primary trigonometric functions, which is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. Its values range from \(-1\) to \(1\).

Understanding the behavior of the sine function across different quadrants is key. In the first and second quadrants, the sine values are positive. However, in the third and fourth quadrants, sine becomes negative.

Since \(285^\circ\) is in the fourth quadrant, \( ext{sin} \, 285^\circ\) is negative. Using the reference angle of \(75^\circ\), we express \( ext{sin} \, 285^\circ\) as \(- ext{sin} \, 75^\circ\). This negative sign arises from the properties of the sine function in the fourth quadrant.

Fourth Quadrant

The fourth quadrant is located between \(270^\circ\) and \(360^\circ\) on the Cartesian coordinate system. In this quadrant, the angle's terminal side lies between the positive x-axis and negative y-axis.

• Sine values are negative since the y-values are negative in the fourth quadrant.
• Cosine values remain positive since they correspond to x-values, which are positive here.

Understanding which trigonometric function is positive or negative in each quadrant helps convert angle values accurately. This knowledge aids in determining the signs of trigonometric function values, like turning \( ext{sin} \, 285^\circ\) into \(- ext{sin} \, 75^\circ\). Knowing that sine is negative here emphasizes the importance of quadrants in trigonometry.

Angle Reduction

Angle reduction involves simplifying an angle to find a related angle that is easier to work with, often using trigonometric properties.

The key step in angle reduction is using reference angles to relate a larger angle to a more manageable acute angle. For angles like \(285^\circ\), we use reference angles to express their trigonometric functions effectively.

To reduce a fourth quadrant angle:

• Find the reference angle by subtracting the angle from \(360^\circ\).
• Determine the function's sign based on the quadrant.
• Use the function's properties (e.g., sine is negative here) to adjust expression.

Thus, to express \(\text{sin} \, 285^\circ\) in terms of an acute angle, we calculate \(360^\circ - 285^\circ = 75^\circ\), then apply sine's negative property in the fourth quadrant, resulting in \(-\text{sin} \, 75^\circ\). This method provides a systematic approach to handling larger angles.