Problem 104 - Precalculus Essentials

Question

In Exercises \(102-105,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. When using the half-angle formulas for trigonometric functions of \(\frac{\alpha}{2},\) I determine the sign based on the quadrant in which \(\alpha\) lies.

Step-by-Step Solution

Answer

The statement does not make sense because the quadrant of \(\frac{\alpha}{2}\) can be different from the quadrant of \(\alpha\). The signs of trigonometric functions of \(\frac{\alpha}{2}\) should be determined based on its own quadrant.

Step 1: Understand the behavior of trigonometric functions

In the unit circle, the sine function is positive in the first and second quadrants, whereas it's negative in the third and fourth quadrants. The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.

Step 2: Apply the concept to the statement

Following the standard behavior, when \(\alpha\) lies in the first or fourth quadrant, cos(\(\alpha\)) is positive, and sin(\(\alpha\)) is positive when \(\alpha\) is in the first or second quadrant. For \(\frac{\alpha}{2}\), its quadrant can be different from \(\alpha\)'s quadrant. Hence, using the quadrant of \(\alpha\) to determine the sign of trigonometric functions for \(\frac{\alpha}{2}\) doesn't make sense.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle helps in understanding the behavior of trigonometric functions. Each point on this circle corresponds to an angle. The x-coordinate of the point where the terminal of an angle intersects the circle gives the cosine value of that angle, while the y-coordinate gives the sine value.

By using the unit circle, we can visualize how angles relate to the sine and cosine functions. Angles on the unit circle are usually measured in radians, but degrees are frequently used as well.

Understanding the unit circle allows you to grasp how the signs of sine and cosine vary across different quadrants. This visualization is particularly useful when dealing with half-angle formulas, where the angle is split into two equal parts.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are key to understanding angles and their relationships within the unit circle. These functions describe the relationships between angles and side lengths in right-angled triangles.

For sine, it is defined as the opposite side over the hypotenuse in a right triangle, while cosine is the adjacent side over the hypotenuse. The tangent function, defined as sine over cosine, gives the ratio of the opposite to adjacent sides.

• The sine function is positive in the first and second quadrants.
• The cosine function is positive in the first and fourth quadrants.

These functions vary systematically around the circle, which is visualized as moving anti-clockwise from the positive x-axis. This systematic behavior can help determine the values of these functions at any angle.

Quadrants

In the coordinate plane, quadrants are the four sections created by dividing the plane with the x-axis and y-axis. They are numbered counterclockwise starting from the top right-hand side.

• Quadrant I: Both x and y coordinates are positive. Therefore, sine, cosine, and tangent are positive.
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive. Sine is positive, while cosine and tangent are negative.
• Quadrant III: Both x and y are negative. In this case, tangent is positive, while sine and cosine are negative.
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative. Cosine is positive, while sine and tangent are negative.

Understanding in which quadrant an angle terminates is essential for determining the signs of trigonometric functions for that angle. However, this doesn't directly apply to half-angle formulas without considering the new quadrant where the resulting half-angle resides.