Problem 63 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator.

Step-by-Step Solution

Answer

The sine of \( -\frac{\pi}{6} \) is \(-\frac{1}{2}\), the cosine is \(\frac{\sqrt{3}}{2}\), and the tangent is \(-\sqrt{3}\).

Step 1: Convert the negative angle to its positive acute counterpart

The negative radian measure \( -\frac{\pi}{6} \) corresponds to moving clockwise on the unit circle from the positive x-axis. After moving clockwise, the amount indicated by \( -\frac{\pi}{6} \), we end up at the same position as moving counter-clockwise \(\frac{11\pi}{6}\). So, let's now consider the angle \(\frac{11\pi}{6}\) which falls in the fourth quadrant.

Step 2: Evaluate the sine, cosine, and tangent

In a unit circle, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Similarly, the cosine of an angle is the x-coordinate. Therefore, the sine of \(\frac{11\pi}{6}\) which is the same as the sine of \( -\frac{\pi}{6} \) is \(-\frac{1}{2}\), and the cosine is \(\frac{\sqrt{3}}{2}\). The tangent, which is the ratio of sine to cosine, is then \(-\frac{1}{\sqrt{3}}\), which simplifies to \(-\sqrt{3}\) when rationalizing the denominator.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1 unit centered at the origin of the coordinate plane. It is a crucial tool in trigonometry because it helps us find the sine, cosine, and tangent of angles. Here's how it works:

• Every point on the unit circle has coordinates \( ( \cos \theta, \sin \theta ) \), where \( \theta \) is the angle formed with the positive x-axis.
• The unit circle represents angles in both degrees and radians, making it versatile.
• Since its radius is 1, the distances on the circle can be easily translated into trigonometric values.

Understanding the unit circle helps us relate geometrical angles to their trigonometric counterparts through the coordinates of points on the circle. This is why the unit circle is a core component of understanding trigonometric functions.

Radian Measure

Radian measure is another way to measure angles, different from degrees. It is based on the radius of a circle and is inherently linked to the unit circle:

• One full revolution around a circle is \( 2\pi \) radians, equivalent to 360 degrees.
• A radian is defined as the angle created when the arc length is equal to the radius.
• Common angles in radians that appear on the unit circle include \( \frac{\pi}{6}, \frac{\pi}{4},\text{ and } \frac{\pi}{3} \).

Radians provide a natural way of working with algebraic expressions involving angles. They make mathematical derivations and calculations smoother because they relate directly to the properties of circles.

Quadrant Angles

Quadrant angles help us identify which area (quadrant) of the unit circle an angle lies in. Understanding this information is essential because trigonometric function signs vary by quadrant:

• The unit circle is divided into four quadrants, each spanning 90 degrees or \( \frac{\pi}{2} \) radians.
• The first quadrant includes angles from 0 to \( \frac{\pi}{2} \), where sine and cosine are both positive.
• In the fourth quadrant, which is where \( \frac{11\pi}{6} \) falls, sine is negative, and cosine is positive.

Identifying the quadrant tells us immediately about the sign of each trigonometric function. This aids in simplifying the calculations, especially when angles can be expressed in multiple equivalent ways, like \( -\frac{\pi}{6} \) being the same as \( \frac{11\pi}{6} \).