Problem 2 - Algebra 2 and Trigonometry

Question

If \(\sin A=\frac{1}{2},\) is \(\cos A=\frac{\sqrt{3}}{2}\) always true? Explain why or why not.

Step-by-Step Solution

Answer

No, because \(\cos A\) can be \(\frac{\sqrt{3}}{2}\) or \(-\frac{\sqrt{3}}{2}\) depending on the quadrant.

Step 1: Understanding the Unit Circle

On the unit circle, every point \(x, y\) corresponds to \(\cos A\) and \(\sin A\) respectively, where \(A\) is the angle formed with the positive x-axis. Since \(\sin A = \frac{1}{2}\), we need to identify angles on the unit circle where the y-coordinate is \(\frac{1}{2}\).

Step 2: Identify Angles with \\(\sin A = \frac{1}{2}\\)

The sine value of \(\frac{1}{2}\) occurs at two specific angles in the standard interval \([0, 2\pi)\): \(A = \frac{\pi}{6} \) and \(A = \frac{5\pi}{6}\). These angles relate to the first and second quadrants of the unit circle, respectively.

Step 3: Evaluate \\(\cos A\\) for Each Angle

For \(A = \frac{\pi}{6}\), the cosine value is \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). However, for \(A = \frac{5\pi}{6}\) in the second quadrant, \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\).

Step 4: Conclusion about \\(\cos A\\)

Since \(\cos A\) can be \( \frac{\sqrt{3}}{2}\) for \(A = \frac{\pi}{6}\) and \(-\frac{\sqrt{3}}{2}\) for \(A = \frac{5\pi}{6}\), \(\cos A = \frac{\sqrt{3}}{2}\) is not always true when \(\sin A = \frac{1}{2}\).

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry that helps visualize the trigonometric functions and their relationships. It is a circle with a radius of 1, centered at the origin of a coordinate plane. In this circle, any angle \(A\) is represented by a point \((x, y)\) on the perimeter, where \(x\) signifies the cosine value and \(y\) signifies the sine value of that angle.

The unit circle effectively allows us to understand how the values of sine and cosine change as the angle increases from 0 to \(2\pi\) radians, encompassing all four quadrants of a coordinate grid. As such, it plays a vital role in defining the fundamental trigonometric identities, as each coordinate reflects the periodic nature of these functions.

• At \(0\) or \(2\pi\), \((1, 0)\): cosine is 1, sine is 0.
• At \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\), \((0, 1)\) or \((0, -1)\): cosine is 0, sine is 1 or -1.
• At \(\pi\), \((-1, 0)\): cosine is -1, sine is 0.

Understanding these points is important to solve problems like checking the truth of expressions involving sine and cosine values.

Sine and Cosine Values

Sine and cosine are trigonometric functions that provide a bridge to understand the relationship between the angles and the ratios of sides in a right triangle. When angles are plotted on the unit circle, the sine of the angle represents the y-coordinate, while the cosine represents the x-coordinate.

For instance, if \(\sin A = \frac{1}{2}\), you must find the angles that result in this y-coordinate along the unit circle. At \(A = \frac{\pi}{6}\) (approximately 30 degrees) and \(A = \frac{5\pi}{6}\) (approximately 150 degrees), the sine of the angle is \(\frac{1}{2}\).

Yet, cosine values differ with these angles:

• \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)
• \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\)

This distinction arises from the nature of angles in different quadrants, which we will explore further. These values demonstrate how sine and cosine depend on both the magnitude of the angle and its location within the unit circle.

Angles in Quadrants

The coordinate plane can be divided into four quadrants, each influencing the sign of sine and cosine values:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both values are negative.
• Quadrant IV: Sine is negative, cosine is positive.

The effect of these quadrant rules determines whether trigonometric identities hold true across different angles.

Considering \(\sin A = \frac{1}{2}\), the angle \(A\) might land in either the first quadrant ( \(\frac{\pi}{6}\)) or the second quadrant (\(\frac{5\pi}{6}\)). In the first quadrant, both sine and cosine values are positive due to the x-y coordinates within that section of the unit circle, which explains \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). Conversely, since the second quadrant affects signs, \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\).

Understanding these rules is essential for verifying or disproving certain trigonometric equations, as each quadrant presents its unique properties that affect the outcome of expressions like \(\cos A = \frac{\sqrt{3}}{2}\).