Problem 109 - Algebra and Trigonometry

Question

Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.

Step-by-Step Solution

Answer

A reference angle is the acute angle made with the x-axis which has the same trigonometric values (except for a possible sign change) as the given angle. In the example of \(330^\circ\), its reference angle is \(30^\circ\). The \( \sin(330^\circ) \) will be equal to \( -\sin(30^\circ) \) and \( \cos(330^\circ) \) will be equal to \( \cos(30^\circ) \) as \(330^\circ\) lies in the fourth quadrant where sine is negative and cosine is positive.

Step 1: Defining Reference Angle

A reference angle is defined as the acute angle formed by the x-axis and the terminal side of the given angle in standard position. In simple terms, it's the smallest angle that the given angle makes with the x-axis.

Step 2: Application of Reference Angles

Reference angles are used in order to simplify the calculation process. Regardless of the quadrant in which the angle lies, its reference angle will have the same trigonometric values, except potentially with a different sign. The sign is determined depending on the quadrant of the original angle.

Step 3: Practical Example

Let's consider an angle of \(330^\circ\). The reference angle for this would be the difference between the angle and \(360^\circ\) if it's located in the fourth quadrant. So, the reference angle here would be \(360^\circ - 330^\circ = 30^\circ\). The sine, cosine and tangent values of the \(30^\circ\) reference angle would be the same as that of the \(330^\circ\) angle, only the signs might vary depending on the quadrant. Since \(330^\circ\) is in the 4th quadrant, where sine is negative and cosine is positive, we have \( \sin(330^\circ) = - \sin(30^\circ) \) and \( \cos(330^\circ) = \cos(30^\circ) \).

Step 4: Conclusion

In the world of trigonometry, reference angles can simplify the process of evaluating functions drastically. They provide a standard measure to jump back to, thereby reducing the complexity of calculations.